% multiplicateurs de Fourier, inconditionnalité, propriétés d'approximation, ensembles Lambda(p), ensembles de Sidon, ensembles de Rosenthal, ensembles d'entiers équidistribués, multiplicateurs de Schur, classes de Schatten

% Fourier multipliers, unconditionality, approximation properties, Lambda(p) sets, Sidon sets, Rosenthal sets, equidistributed integer sets, Schur multipliers, Schatten classes

% 11K31, 42A55, 43A15, 43A46, 46B04, 46B15, 46B28, 47B10, 60G99

% La thèse a été rapportée par Jean Esterle et Françoise Piquard.
% Gilles Godefroy, Daniel Li, Hervé Queffélec et Jean Saint Raymond ont été les membres du jury présidé par par Myriam Déchamps. 





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\def\stefan{Stefan Neuwirth\par
Laboratoire de Math\'ematiques\par
Universit\'e de Franche-Comt\'e\par
16, route de Gray, 25000 Besan\c{c}on\par
neuwirth@math.univ-fcomte.fr\par}

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bet-ween 
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Un-be-dingt-heit
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uni-form-ly
uni-que
}


%\voffset=-10mm\textheight=220.3mm
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%\parindent=0mm
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\makeindex


\author{Stefan Neuwirth}
\title{Multiplicateurs et analyse fonctionnelle}
\date{}

\begin{document}



\parindent=0pt

\thispagestyle{empty}{



\begin{center}

\LARGE\sf

Th\`ese de Doctorat de l'Universit\'e Paris VI

Math\'ematiques

\vfill

Stefan Neuwirth

\vfill

\Huge\sf

 Multiplicateurs et 

analyse fonctionnelle

\end{center}

\vfill\vfill

\LARGE\sf

Th\`ese soutenue le 19 janvier 1999

devant le jury compos\'e de

\vskip6pt



\halign{#\ \hfil&#\hfil\cr
M\raise6pt\hbox{\large\sf me}&Myriam D\'echamps,\cr
M.&Jean Esterle,\cr
M.&Gilles Godefroy,\cr
M.&Daniel Li,\cr
M\raise6pt\hbox{\large\sf me}&Fran\c{c}oise Piquard,\cr
M.&Herv\'e Queff\'elec et\cr
M.&Jean Saint Raymond\cr}

}
\pagebreak\hbox{}\thispagestyle{empty}\vfill\pagebreak

\hbox{}\vfill

{\large\leftskip=9.5mm\rightskip=9.5mm

Cette th\`ese a \'et\'e \'ecrite sous la direction de M.\ Gilles
Godefroy et de M.\ Daniel Li. Elle est redevable de leur culture, de
la pr\'ecision de leur \'ecoute, ainsi que de leurs multiples
questions. \vskip12pt

M\raise5pt\hbox{\scriptsize me} Fran\c{c}oise Piquard et M.\ Jean
Esterle ont accept\'e de lire cette th\`ese et de r\'ediger un
rapport. M\raise5pt\hbox{\scriptsize me} Myriam D\'echamps, M.\
Herv\'e Queff\'elec et M.\ Jean Saint Raymond ont bien voulu faire
partie du jury. Leur pr\'esence m'honore. \vskip12pt

Ce travail scientifique a \'et\'e l'occasion de nombreuses rencontres,
d'abord \`a Paris, puis \`a Columbia dans le Missouri, enfin \`a
Besan\c{c}on. Je les garderai dans mon c\oe ur.

}
\thispagestyle{empty}\vfill\vfill\pagebreak
\hbox{}\thispagestyle{empty}\vfill\pagebreak
\tableofcontents\vfill\pagebreak

\baselineskip=14.2pt minus 1.2pt
\hbox{}\vfill

\markboth{}{}
\begin{center}\thispagestyle{empty}
 {\large\bf R\'esum\'e de la th\`ese}
\addcontentsline{toc}{section}{R\'esum\'e de la th\`ese
% en langue fran\c{c}aise
}

\end{center}

{\leftskip=1cm\rightskip=1cm
Nous \'etudions plusieurs propri\'et\'es fonctionnelles
d'inconditionnalit\'e en les exprimant \`a l'aide de multiplicateurs.
La premi\`ere partie est consacr\'ee \`a l'\'etude de ph\'enom\`enes
d'inconditionnalit\'e isom\'etrique et presqu'iso\-m\'etrique dans les
espaces de Banach s\'eparables. Parmi ceux-ci, la notion la plus
g\'en\'erale est celle de ``propri\'et\'e d'approximation
inconditionnelle m\'etrique''. Nous la caract\'erisons parmi les
espaces de Banach de cotype fini par une propri\'et\'e simple
d'``inconditionnalit\'e par blocs''. En nous ramenant \`a des
multiplicateurs de Fourier, nous \'etudions cette propri\'et\'e dans
les sous-espaces des espaces de Banach de fonctions sur le cercle qui
sont engendr\'es par une suite de ca\-rac\-t\`eres $\e^{\ii nt}$. Nous
\'etudions aussi les suites basiques inconditionnelles iso\-m\'etriques
et presqu'isom\'etriques de caract\`eres, en particulier les ensembles
de Sidon de constante asymptotiquement $1$. Nous obtenons dans chaque
cas des propri\'et\'es combinatoires sur la suite. La propri\'et\'e
suivante des normes ${\fam0 L}^p$ est cruciale pour notre \'etude: si $p$ est
un entier pair,
$\int|f|^p=\int{|f^{p/2}|}^2=\sum|\widehat{f^{p/2}}(n)|{\vphantom{|f^{p/2}|}}^2$
est une
expression polyn\^o\-miale en les coefficients de Fourier de $f$ et
$\bar f$. Nous proposons d'ailleurs une estimation pr\'ecise de la
constante de Sidon des ensembles \`a la Hadamard. La deuxi\`eme partie
\'etudie les multiplicateurs de Schur: nous caract\'erisons les suites
basiques inconditionnelles isom\'etriques d'entr\'ees de matrice
$\e_{ij}$ dans la classe de Schatten $S^p$. Les propri\'et\'es
combinatoires que nous obtenons portent sur les chemins dans le
r\'eseau  
%\hbox{I\kern-2.3pt N$\times$I\kern-2.3pt N} 
$\N\times\N$ \`a sommets dans
cet ensemble. La troisi\`eme partie \'etudie le rapport entre la
croissance d'une suite d'entiers et les propri\'et\'es harmoniques et
fonctionnelles de la suite de caract\`eres associ\'ee. Nous montrons
en particulier que toute suite polyn\^omiale, ainsi que la suite des
nombres premiers, contient un ensemble $\Lambda(p)$ pour tout $p$ qui
n'est pas de Rosenthal.

}

\vfill\vfill\pagebreak\hbox{}\vfill

\markboth{}{}
\begin{center}
 {\large\bf Zusammenfassung der Dissertation}
\addcontentsline{toc}{section}{Zusammenfassung der Dissertation auf deutsch}

\end{center}

{\leftskip=1cm\rightskip=1cm
Verschiedene funktionalanalytische Unbedingtheitseigenschaften werden
mittels Multiplikatoren untersucht. Teil I beschreibt die Begriffe
isometrischer und fast isometrischer Unbedingtheit in separablen
Banachr\"aumen. Am allgemeinsten ist die metrische unbedingte
Approximationseigenschaft gefasst. Wir charakterisieren diese f\"ur
Banach\-r\"aume mit endlichem Kotyp durch eine einfache ``blockweise''
Unbedingtheit. Daraufhin betrachten wir genauer den Fall von
Funktionenr\"aumen auf dem Einheitskreis, die durch eine Folge von
Frequenzen  $\e^{\ii nt}$ aufgespannt werden. Wir untersuchen
isometrisch und fast isometrisch unbedingte Basisfolgen von
Frequenzen, unter anderem Sidonmengen mit einer Konstante 
asymptotisch zu $1$. 
F\"ur jeden Fall erhalten wir kombinatorische Eigenschaften der
Folge. Die folgende Eigenschaft der ${\fam0 L}^p$ Normen ist
entscheidend f\"ur diese Arbeit: Ist $p$ eine gerade Zahl, so ist
$\int|f|^p=\int{|f^{p/2}|}^2=\sum|\widehat{f^{p/2}}(n)|
{\vphantom{|f^{p/2}|}}^2$ ein
polynomialer Ausdruck der Fourierkoeffizienten von $f$ und $\bar
f$. Des weiteren erhalten wir eine genaue Absch\"atzung der
Sidonkonstante von Hadamardfolgen. Teil II untersucht
Schurmultiplikatoren: Wir kennzeichnen die isometrisch unbedingten
Basisfolgen von Matrixkoeffizienten $\e_{ij}$ in der Schattenklasse
$S^p$ durch die Wege auf dem Gitter $\N\times\N$ mit Eckpunkten in dieser
Folge. Teil III befasst sich mit dem Zusammenhang zwischen dem
Wachstum einer Folge von ganzen Zahlen und den harmonischen und
funktionalanalytischen Eigenschaften der zugeh\"origen Folge von
Frequenzen. Wir zeigen insbesondere, dass jede polynomiale Folge,
sowie die Primzahlenfolge, eine Unterfolge enth\"alt, die zwar $\Lambda(p)$
f\"ur jedes $p$ aber keine Rosenthalmenge ist.

}
\vfill\vfill\pagebreak

\hbox{}\vfill

\begin{center}
  {\large\bf Abstract of the thesis}
\addcontentsline{toc}{section}{Abstract of the thesis in English}
\end{center}

{\leftskip=1cm\rightskip=1cm
We study several functional properties of unconditionality and state
them as a property of families of multipliers. This Thesis has three
parts. Part I is devoted to the study of several notions of isometric
and almost isometric unconditionality in separable Banach spaces. The
most general such notion is that of ``metric unconditional
approximation property''. We characterize this ``\umap/'' by a
simple property of ``block unconditionality'' for spaces with
nontrivial cotype. We focus on subspaces of Banach spaces of functions
on the circle spanned by a sequence of characters $\e^{\ii nt}$. There
\umap/ may be stated in terms of Fourier multipliers.  We express
\umap/ as a simple combinatorial property of this sequence. We
obtain a corresponding result for isometric and almost isometric basic
sequences of characters. Our study uses the following crucial property
of the ${\fam0 L}^p$ norm for even $p$:
$\int|f|^p=\int{|f^{p/2}|}^2=\sum|\widehat{f^{p/2}}(n)|{\vphantom{|f^{p/2}|}}^2$
is a
polynomial expression in the Fourier coefficients of $f$ and $\bar f$.
As a byproduct, we get a sharp estimate of the Sidon constant of sets
{\it \`a la}\/ Hadamard. Part II studies Schur multipliers: we
characterize isometric unconditional basic sequences of matrix entries
$\e_{ij}$ in the Schatten class $S^p$. The combinatorial properties
that we obtain concern paths on the lattice $\N\times\N$ with vertices
in this set. Part III studies the relationship between the growth rate
of an integer sequence and harmonic and functional properties of the
corresponding sequence of characters. We show in particular that every
polynomial sequence contains a set that is $\Lambda(p)$ for all $p$
but is not a Rosenthal set. This holds also for the sequence of
primes.

}

\vfill\vfill\pagebreak\thispagestyle{empty}\hbox{}\vfill\pagebreak

\baselineskip=15.2pt minus 1.2pt
\chapter{Introduction}\label{chap0}

\section{Position du probl\`eme}

Cette th\`ese se situe au croisement de l'analyse fonctionnelle et de
l'analyse harmonique.  Nous allons donner des \'el\'ements de
r\'eponse \`a la question g\'en\'erale suivante.

\questf
Quelle est la validit\'e de la repr\'esentation
\begin{equation}\label{intro-these:q1}
f\sim\sum \varrho_q\e^{\ii\vartheta_q}\e_q
\end{equation}
de la fonction $f$
comme s\'erie de fr\'equences $\e_q$ d'intensit\'e $\varrho_q$ et de
phase $\vartheta_q$~?

Les r\'eponses seront donn\'es en termes de l'espace de fonctions
$X\ni f$ et du spectre $E\supseteq\{q:\varrho_q>0\}$.

\subsection{Chapitre \ref{chap1}}

Consid\'erons par exemple les deux questions classiques suivantes dans
le cadre des espaces de Banach homog\`enes de fonctions sur le tore
$\T$, des fr\'equences de Fourier $\e_q(t)=\e^{\ii qt}$ et des
coefficients de Fourier
$$\varrho_q\e^{\ii\vartheta_q}=\int\e_{-q}f=\widehat{f}(q).$$

\questsubf\label{intro:q1}
Est-ce que pour les fonctions $f\in X$ \`a spectre
dans $E$
$$\Bigl\| f-\sum_{|q|\le n}\varrho_q\e^{\ii\vartheta_q}\e_q
\Bigr\|_X\tol_{n\to\infty}0\hbox{ ?}$$
Cela revient \`a demander:
est-ce que la suite $\{\e_q\}_{q\in E}$ rang\'ee par valeur absolue
$|q|$ croissante est une base de $X_E$~?  En d'autres termes, la suite
des multiplicateurs idempotents relatifs $T_n:X_E\to X_E$ d\'efinie
par
$$T_n\e_q=\left\{\begin{array}{cl}
    \e_q&\hbox{si }|q|\le n\\
    0&\hbox{sinon}
\end{array}\right.$$
est-elle uniform\'ement born\'ee sur $n$~? Soit $E=\Z$. Un \'el\'ement
de r\'eponse classique est le suivant. 
%Soit $E=\Z$. La norme des $T_n$ se calcule:
$$
\|T_n\|_{{\fam0 L}^2(\sT)\to{\fam0 L}^2(\sT)}=1\ ,\ 
\|T_n\|_{{\fam0 L}^1(\sT)\to{\fam0 L}^1(\sT)}= 
\|T_n\|_{\script{C}(\sT)\to\script{C}(\sT)}\asymp\log n.
$$
On sait de plus que les $T_n$ sont aussi uniform\'ement born\'es sur
${\fam0 L}^p(\T)$, $1<p<\infty$.

\questsubf 
Est-ce que la somme de la s\'erie $\sum
\varrho_q\e^{\ii\vartheta_q}\e_q$ d\'epend de l'ordre dans lequel on
somme les fr\'equences~? Cette question est \'equivalente \`a la
suivante: la nature de $\sum \varrho_q\e^{\ii\vartheta_q}\e_q$
d\'epend-elle des phases $\vartheta_q$~? En termes fonctionnels,
$\{\e_q\}_{q\in E}$ forme-t-elle une suite basique inconditionnelle
dans $X$~? Cette question s'\'enonce aussi en termes de
multiplicateurs relatifs: la famille des $T_\epsilon:X_E\to X_E$ avec
$$T_\epsilon\e_q=\epsilon_q\e_q\hbox{ et }\epsilon_q=\pm1$$
est-elle
uniform\'ement born\'ee sur les choix de signes $\epsilon$~?  Un
\'el\'ement de r\'eponse classique est le suivant. Soit $E=\Z$. Alors
$$\|T_\epsilon\|_{{\fam0 L}^2(\sT)\to {\fam0 L}^2(\sT)}=1;$$
si $p\ne2$, il existe un
choix de signes $\epsilon$ tel que $T_\epsilon$ n'est pas born\'e sur
${\fam0 L}^p(\T)$.

\questsubf
Peut-on am\'eliorer ce ph\'enom\`ene en restreignant
le spectre $E$~?  Cette question m\`ene \`a l'\'etude des
sous-ensembles lacunaires de $\Z$, et a \'et\'e trait\'ee en d\'etail
par Walter Rudin.

Nous choisissons la notion de multiplicateur relatif comme
dictionnaire entre l'a\-na\-ly\-se harmonique et l'analyse fonctionnelle.
Nous d\'eveloppons une technique pour le calcul de la norme de
familles $\{T_\epsilon\}$ de multiplicateurs relatifs. Celle-ci nous
permet de traiter les questions suivantes.

\questsubf\label{intro:q4} 
Est-ce que la norme de $f\in X_E$ d\'epend seulement
de l'intensit\'e $\varrho_q$ de ses fr\'equences $\e_q$, et non pas de
leur phase $\vartheta_q$~? Cela revient \`a demander si $\{\e_q\}_{q\in
  E}$ est une suite basique $1$-inconditionnelle complexe dans $X$.

\questsubf\label{intro:q5}
Est-ce que l'on a pour tout choix de signes ``r\'eel''
$\pm$
$$
\Bigl\|\sum_{q\in E}\pm a_q\e_q\Bigr\|_X= \Bigl\|\sum _{q\in E}
a_q\e_q\Bigr\|_X\hbox{ ?}
$$
En d'autres mots, est-ce que $\{\e_q\}_{q\in E}$ est une suite
basique $1$-inconditionnelle r\'eelle dans $X$~?

La r\'eponse est d\'ecevante dans le cas des espaces ${\fam0 L}^p(\T)$, $p$
non entier pair: seules les fonctions dont le spectre a au plus deux
\'el\'ements v\'erifient ces deux propri\'et\'es. Pour mieux cerner le
ph\'enom\`ene, nous proposons d'introduire la question
presqu'iso\-m\'e\-tri\-que suivante.

\questsubf\label{intro:q6}
Est-ce que la norme de $f\in X_E$ d\'epend
arbitrairement peu de la phase $\vartheta_q$ de ses fr\'equences
$\e_q$~? De mani\`ere pr\'ecise, dans quel cas existe-t-il, pour
chaque $\eps>0$, un sous-ensemble $F\se E$ fini tel que
$$\Bigl\| \sum_{q\in E\setminus F}\varrho_q\e^{\ii\vartheta_q}\e_q
\Bigr\|_X\le(1+\eps)\Bigl\| \sum_{q\in E\setminus F}\varrho_q\e_q
\Bigr\|_X\hbox{ ?}
$$
Dans le cas $X=\script{C}(\T)$, cela signifiera que $E$ est un
ensemble de constante de Sidon ``asymptotiquement $1$''.  De m\^eme,
peut-on choisir pour chaque $\eps>0$ un ensemble fini $F$ tel que pour
tout choix de signe ``r\'eel'' $\pm$
$$
\Bigl\| \sum_{q\in E\setminus F}\pm a_q\e_q
\Bigr\|_X\le(1+\eps)\Bigl\| \sum_{q\in E\setminus F}a_q\e_q
\Bigr\|_X\hbox{ ?}
$$

Toutes ces questions s'agr\`egent autour d'un fait bien connu: sommer
la s\'erie de Fourier de $f$ est une tr\`es mauvaise mani\`ere
d'approcher la fonction $f$ d\`es que l'erreur consid\'er\'ee n'est
pas quadratique. On sait qu'il est alors utile de rechercher des
m\'ethodes de sommation plus lisses, c'est-\`a-dire d'autres suites
approximantes plus r\'eguli\`eres. Il s'agit l\`a de suites
d'op\'erateurs de rang fini sur $X_E$ qui approchent ponctuellement
l'identit\'e de $X_E$.  Nous pourrons toujours supposer que ces
op\'erateurs sont des multiplicateurs. Une premi\`ere question est la
suivante.

\questsubf\label{intro:q7} \
Existe-t-il une suite approximante $\{T_n\}$ de
multiplicateurs idempotents~? Cela revient \`a demander: existe-t-il
une d\'ecomposition de $X_E$ en sous-espaces $X_{E_k}$ de dimension
finie avec
\begin{equation}\label{intro:fdd}
X_E=\bigoplus X_{E_k}\quad\hbox{et}\quad A_k:X_E\to X_{E_k}\ ,\ \e_q\mapsto
\left\{\begin{array}{cl}
    \e_q&\hbox{si }q\in E_k\\
    0&\hbox{sinon}
\end{array}\right.
\end{equation}
telle que la suite des $T_n=A_1+\dots+A_n$ est uniform\'ement
born\'ee sur $n$~?  Soit $E=\Z$. Alors la r\'eponse est identique \`a
la r\'eponse de la question \ref{intro:q1}.

Mais nous pouvons produire dans ce cadre plus g\'en\'eral des
d\'ecompositions inconditionnelles de $X_E$ en r\'eponse \`a la
question suivante.

\questsubf
Pour quels espaces $X$ et spectres $E$ existe-t-il une
d\'ecomposition comme ci-dessus telle que la famille des multiplicateurs
\begin{equation}\label{intro:fddi}
\sum_{k=1}^n\epsilon_kA_k\quad\hbox{avec }
n\ge1\hbox{ et }\epsilon_k=\pm 1
\end{equation}
est uniform\'ement born\'ee~?  Littlewood et Paley ont
montr\'e que la partition de $\Z$ en $\Z=\bigcup E_k$ avec $E_0=\{0\}$ et
$E_k=\{j:2^{k-1}\le |j|<2^k\}$ donne une d\'ecomposition
inconditionnelle des espaces ${\fam0 L}^p(\T)$ avec $1<p<\infty$. D'apr\`es la
r\'eponse \`a la question \ref{intro:q7}, ce n'est pas
le cas \af des espaces ${\fam0 L}^1(\T)$ et $\script{C}(\T)$. 
%Il est utile de noter que l'espace de Hardy $H^1(\T)$ n'admet pas non
%plus de telle d\'ecomposition inconditionnelle quoiqu'il ait une base
%inconditionnelle.  
Une \'etude fine de telles partitions a \'et\'e entreprise par Kathryn
Hare et Ivo Klemes.

Notre technique permet de traiter la question suivante.

\questsubf\label{intro:q9} 
Pour quels espaces $X$ et spectres $E$ existe-t-il une
d\'ecomposition du type \Ref{intro:fdd} telle que 
$$
\Bigl\|\sum\epsilon_kA_kf\Bigr\|_X=\|f\|_X \hbox{ pour
  tout choix de signes }\epsilon_k\hbox{ ?}
$$
La r\'eponse d\'ependra de la nature du choix de signes, qui peut \^etre r\'eel ou
complexe. 

Il est instructif de noter que l'espace de Hardy $H^1(\T)$ n'admet pas de
d\'ecomposition du type \Ref{intro:fdd}. $H^1(\T)$ admet n\'eanmoins
des suites approximantes de multiplicateurs et il existe m\^eme des
suites approximantes de multiplicateurs inconditionnelles au sens o\`u
la famille \Ref{intro:fddi} est uniform\'ememt born\'ee. Cela motive
la question suivante, qui est la plus g\'en\'erale dans notre
contexte.
%Nous traitons finalement la question la plus g\'en\'erale.

\questsubf\label{intro:q10}
Quels sont les espaces $X$ et spectres $E$ tels que pour chaque
$\eps>0$ il existe une suite approximante $\{T_n\}$ sur $X_E$ telle
que
$$
\sup_{\hbox{\scriptsize signes }\epsilon_n}
\Bigl\|\sum\epsilon_n(T_n-T_{n-1})\Bigr\|_X\le1+\eps
$$
En termes fonctionnels, $X_E$ a-t-il la propri\'et\'e d'approximation
inconditionnelle m\'etrique~? Il faudra distinguer le cas des signes
complexes et r\'eels.

\subsection{Chapitre \ref{chap2}}

Nous montrons que notre technique de calcul s'applique {\it mutatis
  mutandis}\/ aux
multiplicateurs de Schur. La re\-pr\'e\-sen\-ta\-tion \Ref{intro-these:q1} est alors la
repr\'esentation matricielle: on note $\e_q=\e_{rc}$ l'entr\'ee de
  matrice en $q=(r,c)\in\N\times\N$, c'est-\`a-dire l'op\'erateur sur
$\ell_2$ qui envoie son $c$i\`eme vecteur de base sur son $r$i\`eme
vecteur de base et de matrice $(\delta_k^r\delta_l^c)_{k,l\ge0}$. On
consid\`ere donc la validit\'e de 
$$
x\sim\sum \varrho_q\e^{\ii\vartheta_q}\e_q\hbox{ avec les coefficients de matrice }
x_q=\varrho_q\e^{\ii\vartheta_q}=\trace\e_q^*x
$$
pour $x$ op\'erateur sur $\ell_2$. Soit $p\ge1$ et
$I\se\N\times\N$. Notre \'etude se fera en termes de la classe de
Schatten $S^p$ de $x$ et du support $I$ de la matrice $(x_q)$
associ\'ee \`a $x$. Nous dirons que $x$ est \`a entr\'ees dans $I$ si
$I\supseteq\{q:x_q\ne0\}$.

%Nous traitons la question suivante: 
\questsubf \label{q:schur} Est-ce que la norme de $x\in S^p$ \`a
entr\'ees dans $I$ d\'epend seulement du module $\varrho_q$ de ses
coefficients de matrice et non pas de leur argument $\vartheta_q$~?
Cela revient \`a demander: quelles sont les suites basiques
d'entr\'ees $1$-in\-con\-di\-tion\-nel\-les dans $S^p$~? En termes de
multiplicateurs de Schur, la question se pose ainsi. Pour quels
ensembles d'entr\'ees $I$ les
$$
T_\epsilon:S^p_I\to S^p_I\ ,\ \e_q\mapsto\epsilon_q\e_q \hbox{
~  avec } |\epsilon_q|=1
$$
sont-ils tous des isom\'etries~?

\subsection{Chapitre \ref{chap3}} 

Nous \'etudions le rapport entre la croissance d'une suite
$\{n_k\}=E\se\Z$
et deux de ses propri\'et\'es harmoniques et fonctionnelles
\'eventuelles, \ie \\
\bloc 
toute fonction int\'egrable \`a spectre dans $E$ est en fait
$p$-int\'egrable pour tout $p<\infty$: $E$ est un ensemble
$\Lambda(p)$ pour tout $p$;\\
\bloc 
toute fonction mesurable born\'ee \`a spectre dans $E$ est en fait continue
\`a un ensemble de mesure nulle pr\`es: $E$ est un ensemble de
Rosenthal.

Nous sommes en mesure de dresser le tableau suivant selon la
croissance\\
\bloc 
polyn\^omiale: $n_k\preccurlyeq k^d$ pour un $d<\infty$, \\
\bloc 
surpolyn\^omiale: $n_k\gg k^d$ pour tout $d\ge1$, \\
\bloc 
sous-exponentielle: $\log n_k\ll k$, \\
\bloc 
g\'eom\'etrique: $\liminf |n_{k+1}/n_k|>1$.

\vskip2ex

\hfill\vbox{\halign{\lower4pt\vbox to 14pt{}\hfil#\ \vrule\hbox to 1pt{}&\vrule\ 
  \hfil#\hfil\ &\vrule\ \hfil#\hfil\ &\vrule\ \hfil#\hfil\cr
  croissance&polyn\^omiale&surpolyn\^omiale\&sous-exponentielle
&g\'eom\'etrique\cr
\noalign{\hrule}\noalign{\vskip1pt\hrule}
  $E$ $\Lambda(p)$ $\forall p$& non& presque
  toujours&oui\cr\noalign{\hrule} $E$ Rosenthal&presque
  jamais\span\omit&oui\cr}}\hfill\hbox{}

\hfill\refstepcounter{thmsubf}\label{intro:table}
Tableau \thethmsubf\hfill\hbox{}



\vskip1ex

Li montre qu'effectivement il existe un ensemble $\Lambda(p)$ pour
tout $p$ qui n'est pas de Rosenthal. Nous traitons les deux questions
suivantes. 

\questsubf 
%Nous montrons que l
Le sch\'ema ci-dessus reste-t-il valable si on consid\`ere \`a la
place de l'ensemble des sous-ensembles $E$ de $\Z$ l'ensemble des
sous-ensembles $E$ d'une suite \`a croissance polyn\^omiale~?

\questsubf\label{intro:random:q2}
% ou aussi de la suite des nombres premiers. En particulier, il existe \`a
Si $E$ n'est pas un ensemble de Rosenthal, $E$ contient-il un ensemble
\`a la fois $\Lambda(p)$ pour tout $p$ et non Rosenthal~?

\section{Inconditionnalit\'e m\'etrique en analyse de Fourier}

Nous r\'epondons dans ce chapitre aux questions \ref{intro:q4},
\ref{intro:q5}, \ref{intro:q6}, \ref{intro:q9} et \ref{intro:q10}.

Comme ces questions distinguent les choix de signe r\'eel et complexe,
nous proposons pour la fluidit\'e de l'expos\'e de fixer un choix de
signes $\U$ qui sera $\U=\T=\{\epsilon\in\C:|\epsilon|=1\}$ dans le
cas complexe et $\U=\D=\{-1,1\}$ dans le cas r\'eel.

\subsection{Propri\'et\'e d'approximation inconditionnelle
  m\'e\-tri\-que}

Seule la question \ref{intro:q10} n'impose pas au pr\'ealable de forme
particuli\`ere \`a la suite de multiplicateurs qui est cens\'ee
r\'ealiser la propri\'et\'e consid\'er\'ee. Afin d'\'etablir un lien
entre la \umap/ et la structure du spectre $E$,
nous faisons le d\'etour par une \'etude g\'en\'erale de cette
propri\'et\'e dans le cadre des espaces de Banach s\'eparables.

\subsubsection{Amorce et queue d'un espace de Banach}

Peter G.\ Casazza et Nigel J.\ Kalton ont d\'ecouvert le crit\`ere suivant:
\begin{prpsubf}
Soit $X$ un espace de Banach s\'eparable. $X$ a la \umap/ si et
seulement s'il existe une suite approximante $\{T_k\}$ telle que 
$$
\sup_{\epsilon\in\sU}\|T_k+\epsilon(\Id-T_k)\|_{\script{L}(X)}
\toll{k\to\infty}1.
$$
\end{prpsubf}
Ceci exprime que la constante d'inconditionnalit\'e entre l'amorce
$T_kX$ et la queue $(\Id-T_k)X$ de l'espace $X$
s'am\'eliore asymptotiquement jusqu'\`a l'optimum pour $k\to\infty$.

La \umap/ s'exprime de mani\`ere plus \'el\'ementaire encore si l'on
choisit d'autres notions adapt\'ees d'amorce et de queue. Nous
proposons en particulier la d\'efinition suivante.
\begin{dfnsubf}
Soit $\tau$ une topologie d'espace vectoriel topologique sur $X$. $X$
a la propri\'et\'e $(u(\tau))$ de $\tau$-inconditionnalit\'e si pour
chaque $x\in X$ et toute suite born\'ee $\{y_j\}$ $\tau$-nulle
l'oscillation 
$$
\osc_{\epsilon\in\sU}\|x+\epsilon y_j\|_X=
\sup_{\delta,\epsilon\in\sU}
\bigl(\|x+\epsilon y_j\|-\|x+\delta y_j\|\bigr)
$$
forme elle-m\^eme une suite nulle.
\end{dfnsubf}

Nous avons alors le th\'eor\`eme suivant.
\begin{thmsubf}\label{intro:THM}
Soit $X$ un espace de Banach s\'eparable de cotype fini avec la
propri\'et\'e $(u(\tau))$. Si $X$ admet une suite approximante
$\{T_k\}$ inconditionnelle et commutative 
telle que $T_kx\mathop{\to}\limits^\tau x$ uniform\'ement
sur la boule unit\'e $B_X$, alors des combinaisons convexes
successives $\{U_j\}$ de $\{T_k\}$ r\'ealisent la \umap/.
\end{thmsubf}
{\it Esquisse de preuve.\/ }
On construit ces combinaisons convexes successives par le biais de
d\'ecompositions skipped blocking. En effet, la propri\'et\'e
$(u(\tau))$ a l'effet suivant sur $\{T_k\}$. Pour chaque $\eps>0$, il
existe une sous-suite $\{S_k=T_{n_k}\}$ telle que toute suite de blocs
$S_{b_k}-S_{a_k}$ obtenue en sautant les blocs $S_{a_{k+1}}-S_{b_k}$
se somme de mani\`ere $(1+\eps)$-inconditionnelle.\\
Soit $n\ge1$. Pour chaque $j$, $1\le j\le n$, la suite de blocs
obtenue en sautant $S_{kn+j}-S_{kn+j-1}$ pour $k\ge0$ est
$(1+\eps)$-inconditionnelle. Il s'agit alors d'estimer la moyenne sur
$j$ de ces suites de blocs. On obtient une suite approximante et
l'hypoth\`ese de cotype fini permet de contr\^oler l'apport des blocs
saut\'es.\\
Alors $X$ a la \umap/ parce que $n$ et $\eps$ sont arbitraires.\eck

\subsubsection{Amorce et queue en termes de spectre de Fourier}

Lorsqu'on consid\`ere l'espace invariant par translation $X_E$, une
amorce et une queue naturelle sont les espaces $X_F$ et $X_{E\setminus
  G}$ pour $F$ et $G$ des sous-ensembles finis de $E$. Nous avons
concr\`etement le lemme suivant.
\begin{lemsubf}
$X_E$ a $(u(\tau_f))$, o\`u $\tau_f$ est la topologie
$$ f_n\mathop{\to}\limits^{\tau_f}0\ \Ssi\ \forall k\ \widehat{f_n}(k)\to0 $$
de convergence simple des coefficients de Fourier, si et seulement si
$E$ est bloc-inconditionnel dans $X$ au sens suivant: quels que soient
$\eps>0$ et $F\se E$ fini, il existe $G\se E$ fini tel que pour
$f\in B_{X_F}$ et $g\in B_{X_{E\setminus G}}$
$$\osc_{\epsilon\in\sU}\|f+\epsilon g\|_X=
\sup_{\delta,\epsilon\in\sU}
\bigl(\|f+\epsilon g\|-\|f+\delta g\|\bigr)\le\eps.$$
\end{lemsubf}

Le th\'eor\`eme \ref{intro:THM}  s'\'enonce donc ainsi dans 
ce contexte particulier.
\begin{thmsubf}\label{intro:bloc}
Soit $E\se\Z$ et $X$ un espace de Banach homog\`ene de fonctions sur
le tore $\T$. Si $X_E$ a la \umap/, alors $E$ est bloc-inconditionnel
dans $X$. Inversement, si $E$ est bloc-inconditionnel dans $X$ et de
plus $X_E$ a la propri\'et\'e d'approximation inconditionnelle et un
cotype fini, alors $X_E$ a la \umap/. En particulier, on a\\
\bm{i} Soit $1<p<\infty$. ${\fam0 L}^p_E(\T)$ a la \umap/ si et seulement si
$E$ est bloc-inconditionnel dans ${\fam0 L}^p(\T)$.\\
\bm{ii} ${\fam0 L}^1_E(\T)$ a la \umap/ si et seulement si ${\fam0 L}^1_E(\T)$ a la
propri\'et\'e d'approximation inconditionnelle et $E$ est
bloc-inconditionnel dans ${\fam0 L}^1(\T)$.\\ 
\bm{iii} Si $E$ est bloc-inconditionnel dans $\script{C}(\T)$ et $E$
est un ensemble de Sidon, alors $\script{C}_E(\T)$ a la \umap/.
\end{thmsubf}

Donnons une application de ce th\'eor\`eme.
\begin{prpsubf}
Soit $E=\{n_k\}\se\Z$. Si $n_{k+1}/n_k$ est un entier impair pour
tout $k$, alors $\script{C}_E(\T)$ a la \umap/ r\'eelle.
\end{prpsubf}
{\it Preuve.\/ }
Comme $E$ est n\'ecessairement un ensemble de Sidon, il suffit de
v\'erifier que $E$ est bloc-inconditionnel. Soient
$\eps>0$ et $F\se E\cap[-n,n]$. Soit $l$ tel que $|n_l|\ge \pi n/\eps$
et $G=\{\lst n{l-1}\}$. Soit $f\in B_{\script{C}_F}$ et $g\in
B_{\script{C}_{E\setminus G}}$. Alors $g(t+\pi/n_l)=-g(t)$ par
hypoth\`ese et 
$$ |f(t+\pi/n_l)-f(t)|\le \pi/|n_l|\cdot\|f'\|_\infty\le\pi
n/|n_l|\le\eps $$
par l'in\'egalit\'e de Bernstein. Alors, pour un certain $u\in\T$
\begin{eqnarray*}
\|f-g\|_\infty&=&|f(u)+g(u+\pi/n_l)|\\
&\le&|f(u+\pi/n_l)+g(u+\pi/n_l)|+\eps\\
&\le&\|f+g\|_\infty+\eps.
\end{eqnarray*}
Donc $E$ est bloc-inconditionnel au sens r\'eel.\eck

En particulier, soit la suite g\'eom\'etrique $G=\{3^k\}$. Alors
$\script{C}_G(\T)$ et $\script{C}_{G\cup-G}(\T)$ ont la \umap/
r\'eelle.

\questsubf\label{intro:q2.1}
Qu'en est-il de la \umap/ complexe et qu'en est-il de la suite
g\'eom\'etrique $G=\{2^k\}$~?
 
\subsection{Norme de multiplicateurs et conditions
  combinatoires}\label{intro:nmcc}

Nous proposons ici une m\'ethode uniforme pour r\'epondre aux
questions \ref{intro:q4}, \ref{intro:q5}, \ref{intro:q6},
\ref{intro:q9} et \ref{intro:q10}. En effet, les questions
\ref{intro:q4}, \ref{intro:q5} et \ref{intro:q6} reviennent \`a
\'evaluer l'oscillation de la norme 
$$
\Theta(\epsilon,a)=
\|\epsilon_0a_0\e_{r_0}+\dots+\epsilon_ma_m\e_{r_m}\|_X.
$$
La question \ref{intro:q9} revient \`a \'evaluer l'oscillation de la
norme 
\begin{eqnarray*}
\Psi(\epsilon,a)&=&\Theta((\overbrace{1,\dots,1}^j,
\overbrace{\epsilon,\dots,\epsilon}^{m-j}),a)\\
&=&\|a_0\e_{r_0}+\dots+a_j\e_{r_j}+\epsilon a_{j+1}\e_{r_{j+1}}+\dots+
\epsilon a_m\e_{r_m}\|_X
\end{eqnarray*}
Par le th\'eor\`eme \ref{intro:bloc}, la question \ref{intro:q10}
revient \`a \'etudier cette m\^eme expression dans le cas particulier
o\`u on fait un saut de grandeur arbitraire entre $r_j$ et
$r_{j+1}$.\\
Dans le cas des espaces $X={\fam0 L}^p(\T)$, $p$ entier pair, ces normes sont
des polyn\^omes en $\epsilon$, $\epsilon^{-1}$, $a$ et $\bar a$. Dans
le cas des espaces $X={\fam0 L}^p(\T)$, $p$ non entier pair, elles s'expriment
comme des s\'eries. Il n'y a pas moyen d'exprimer ces normes comme
fonction $\script{C}^\infty$ pour $X=\script{C}(\T)$.

Soit $X={\fam0 L}^p(\T)$. D\'eveloppons $\Theta(\epsilon,a)$. Posons
$q_i=r_i-r_0$. On peut supposer $\epsilon_0=1$ et $a_0=1$. Nous
utilisons la notation suivante:
$$
{x\choose\alpha}=
\frac{x(x-1)\cdots(x-n+1)}{\alpha_1!\alpha_2!\dots}\quad\hbox{pour
  }\alpha\in\N^m\hbox{ tel que }\sum\alpha_i=n
$$
Alors, si $|a_1|,\dots,|a_m|<1/m$ lorsque $p$ n'est pas un entier pair
et sans restriction sinon, 
\begin{eqnarray}
\nonumber
\Theta(\epsilon,a)
&=&\int\biggl|\sum_{n\ge0}\bip{p/2}n
\biggl(\sum_{i=1}^m\epsilon_ia_i\e_{q_i}
\biggr)^n\biggr|^2\\
\nonumber&=&\int\biggl|
\sum_{n\ge0}\bip{p/2}n
\sum_{\scriptstyle\alpha:\lst\alpha m\ge0
 \atop\scriptstyle\sm\alpha m=n}
\bi n\alpha
\epsilon^\alpha a^\alpha
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}
\biggr|^2\\
\nonumber&=&\int\biggl|\sum_{\alpha\in\sN^m}
\bip{p/2}\alpha
\epsilon^\alpha a^\alpha
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}\biggr|^2\\
\nonumber&=&\sum_{R\in\script{R}}
\biggl|\sum_{\alpha\in R}
\bip{p/2}\alpha\epsilon^\alpha a^\alpha
\biggr|^2\\
\nonumber&=&
\sum_{\alpha\in\sN^m}
{\bip{p/2}\alpha}^2|a|^{2\alpha}+
\sum_{\scriptstyle\alpha\ne\beta\in\sN^m\atop\scriptstyle\alpha\sim\beta}
\bip{p/2}\alpha\bip{p/2}\beta
\epsilon^{\alpha-\beta}a^\alpha \bar a^\beta\label{intro:gc:res}
\end{eqnarray}
o\`u $\script{R}$ est la partition de $\N^m$ induite par la relation
d'\'equivalence 
$$\alpha\sim\beta\ssi\sum\alpha_iq_i=\sum\beta_iq_i.$$ 

Nous pouvons r\'epondre imm\'ediatement aux questions \ref{intro:q4}
et \ref{intro:q5} pour $X={\fam0 L}^p(\T)$. 


\subsubsection[Suites basiques $1$-inconditionnelles complexes]
{Question \ref{intro:q4}: suites basiques $1$-inconditionnelles complexes}
\label{sec:intro:q4}

Soient $r_0,\dots r_m$ sont choisis dans $E$, alors \Ref{intro:gc:res}
doit \^etre constante pour $a\in\{|z|<1/m\}^m$ et $\epsilon\in\T^m$. Cela
veut dire que pour tous $\alpha\ne\beta\in\N^m$,
$$
\sum\alpha_iq_i\ne\sum\beta_iq_i\quad\hbox{ou}\quad
\bip{p/2}\alpha\bip{p/2}\beta=0.
$$

\bloc
Si $p$ n'est pas un entier pair, alors
$\bip{p/2}\alpha\bip{p/2}\beta\ne0$ pour tous $\alpha,\beta\in\N^m$ et
on a les relations arithm\'etiques suivantes sur $q_1,q_2,0$:
\begin{eqnarray*}
\overbrace{q_1+\dots+q_1}^{|q_2|}=
\overbrace{q_2+\dots+q_2}^{|q_1|}&&\mbox{si }q_1q_2>0;\\
\overbrace{q_1+\dots+q_1}^{|q_2|}+\overbrace{q_2+\dots+q_2}^{|q_1|}
=0&&\mbox{sinon.}
\end{eqnarray*}
Il suffit donc de prendre $\alpha=(|q_2|,0,\dots)$,
$\beta=(|q_1|,0,\dots)$ et $\alpha=(|q_2|,|q_1|,0,\dots)$,
$\beta=(0,\dots)$ respectivement pour conclure que $\{r_0,r_1,r_2\}$
n'est pas une suite basique $1$-inconditionnelle complexe dans ${\fam0
  L}^p(\T)$ si $p$ n'est pas un entier pair.

\bloc
Si $p$ est un entier pair, $\bip{p/2}\alpha\bip{p/2}\beta=0$ si et
seulement si 
$$\sum\alpha_i>p/2\quad\hbox{ou}\quad\sum\beta_i>p/2.$$ 
On obtient que
$E$ est une suite basique $1$-inconditionnelle dans ${\fam0 L}^p(\T)$ si et
seulement si $E$ est ``$p$-ind\'ependant'', c'est-\`a-dire que
$\sum\alpha_i(r_i-r_0)\ne\sum\beta_i(r_i-r_0)$ pour tous
$r_0,\dots,r_m\in E$ et $\alpha\ne\beta\in\N^m$ tels que
$\sum\alpha_i,\sum\beta_i\le p/2$. Cette condition est \'equivalente
\`a: tout entier $n\in\Z$ s'\'ecrit de mani\`ere au plus unique comme
somme de $p/2$ \'el\'ements de $E$.


\subsubsection[Suites basiques $1$-inconditionnelles r\'eelles]
{Question \ref{intro:q5}: suites basiques $1$-inconditionnelles r\'eelles}

Les suites basiques $1$-inconditionnelles r\'eelles et complexes
co\"\i ncident et la r\'eponse \`a la question \ref{intro:q5} est
identique \`a la r\'eponse \`a la question \ref{intro:q4}. En effet,
d\`es qu'une relation arithm\'etique $\sum(\alpha_i-\beta_i)q_i$
p\`ese sur $E$, on peut supposer que $\alpha_i-\beta_i$ est impair
pour au moins un $i$ en simplifiant la relation par le plus grand
diviseur commun des $\alpha_i-\beta_i$. Mais alors \Ref{intro:gc:res}
n'est pas une fonction constante pour $\epsilon_i$ r\'eel.

Cette propri\'et\'e est propre au tore $\T$. En effet, par exemple la
suite des fonctions de Rademacher est $1$-inconditionnelle r\'eelle
dans $\script{C}(\D^\infty)$, alors que sa constante
d'inconditionnalit\'e complexe est $\pi/2$.


\subsubsection[Suites basiques inconditionnelles m\'etriques]
{Question \ref{intro:q6}: suites basiques inconditionnelles m\'etriques}

On peut m\^eme tirer des cons\'equences utiles du calcul de
\Ref{intro:gc:res} dans le cas pres\-qu'iso\-m\'e\-tri\-que. Il faut pour cela
prendre la pr\'ecaution suivante qui permet un passage \`a la
limite. Soit $0<\varrho<1/m$. Alors 
$$\big\{\Theta\colon\U^m\times\{|z|\le\varrho\}^m\to\R^+:q_1,\dots,
q_m\in\Z^m\big\}$$
est un sous-ensemble relativement compact de
$\script{C}^\infty(\U^m\times\{|z|\le\varrho\}^m)$. Il en d\'ecoule
que si $E$ est une suite basique inconditionnelle m\'etrique, alors
certains coefficients de \Ref{intro:gc:res} deviennent arbitrairement
petits lorsque $q_1,\dots,q_m$ sont choisis grands.

Donnons deux cons\'equences de ce raisonnement.
\begin{prpsubf}
Soit $E\se\Z$.\\
\bm{i}
Soit $p$ un entier pair. Si $E$ est une suite basique inconditionnelle
m\'etrique r\'eelle, alors $E$ est en fait une suite basique
$1$-inconditionnelle complexe \`a un ensemble fini pr\`es.\\
\bm{ii}
Si $E$ est un ensemble de Sidon de constante asymptotiquement $1$,
alors 
$$\XE=\sup_{G\se E\hbox{\scriptsize\ fini}}\,
\inf\bigl\{|\zeta_1p_1+\dots+\zeta_mp_m|:
\lst p m\in E\setminus G\hbox{ distincts}\bigr\}>0$$
pour tout $m\ge1$ et $\zeta\in{\Z^*}^m$.
\end{prpsubf}
On peut exprimer cette derni\`ere propri\'et\'e en disant que la
relation arithm\'etique $\zeta$ ne persiste pas sur $E$.


\subsubsection[Propri\'et\'e d'approximation inconditionnelle m\'etrique]
{Question \ref{intro:q10}: propri\'et\'e d'approximation inconditionnelle m\'e\-tri\-que}

On peut appliquer la technique du paragraphe pr\'ec\'edent en
observant que si $X_E$ a la \umap/, alors
$$
\osc_{\epsilon\in\sU}\Psi(\epsilon,a)
\toll{\lstp r{j+1}m\in E\to\infty}0.
$$

\begin{dfnsubf}
$E$ a la propri\'et\'e $\J{n}$ de bloc-ind\'ependance si pour tout
$F\se E$ fini il existe $G\se E$ fini tel que si un $k\in\Z$ admet
deux repr\'esentations comme somme de $n$ \'el\'ements de
$F\cup(E\setminus G)$
$$\sm pn=k=\sm{p'}n,$$ 
alors
$$\mes{\{j:p_j\in F\}}\quad\hbox{et}\quad\mes{\{j:p'_j\in F\}}$$
sont \'egaux \(choix de signes complexe $\U=\T$\) ou de m\^eme parit\'e
\(choix de signes r\'eel $\U=\D$\).
\end{dfnsubf}
\begin{thmsubf}
Soit $E\se\Z$.\\
\bm{i}
Si $X={\fam0 L}^p(\T)$, $p$ entier pair, alors ${\fam0 L}^p_E(\T)$ a la \umap/ si et
seulement si $E$ satisfait $\J{p/2}$.\\
\bm{ii}
Si $X={\fam0 L}^p(\T)$, $p$ non entier pair, ou $X=\script{C}(\T)$, alors
$X_E$ a la \umap/ seulement si $E$ satisfait
$$\XE=\sup_{G\se E\hbox{\scriptsize\ fini}}\,
\inf\bigl\{|\zeta_1p_1+\dots+\zeta_mp_m|:
\lst p m\in E\setminus G\hbox{ distincts}\bigr\}>0$$
pour tout $m\ge1$ et $\zeta\in{\Z^*}^m$ tel que $\sum\zeta_i$ est non
nul \(cas complexe\) ou impair \(cas r\'eel\).
\end{thmsubf}

On obtient la hi\'erarchie suivante.
$$
\parbox{13mm}{$\SCE$ a\\\umap/}\imp
\parbox{25mm}{$\SLPE$ a \umap/,\\$p$ non entier pair}
\imp\dots\imp
\parbox{15mm}{$\SLE{2n+2}$\\ a \umap/}\imp
\parbox{14mm}{$\SLE{2n}$ a\\\umap/}
\imp\dots\imp\parbox{13mm}{$\SLE{2}$ a\\\umap/.}
$$

Nous pouvons r\'epondre \`a la question \ref{intro:q2.1}. Soit
$G=\{j^k\}$ avec $j\in\Z\setminus\{-1,0,1\}$ et consid\'erons
$\zeta=(j,-1)$. Alors $\langle\zeta,G\rangle=0$. Donc
$\script{C}_G(\T)$ n'a pas la \umap/ complexe. $\script{C}_G(\T)$ n'a
pas la \umap/ r\'eelle si $j$ est pair.


\subsubsection{Deux exemples}

\`A l'aide de nos conditions arithm\'etiques, nous sommes \`a m\^eme
de prouver la proposition suivante.
\begin{prpsubf}
Soit $\sigma>1$ et $E$ la suite des parties enti\`eres de
$\sigma^k$. Alors les assertions suivantes sont \'equivalentes.\\
\bm{i}
$\sigma$ est un nombre transcendant.\\
\bm{ii}
${\fam0 L}^p_E(\T)$ a la \umap/ complexe pour tout $p$ entier pair.\\
\bm{iii}
$E$ est une suite basique inconditionnelle m\'etrique dans chaque
${\fam0 L}^p(\T)$, $p$ entier pair.\\
\bm{iv} Pour chaque $m$ donn\'e, la constante de Sidon des sous-ensembles 
\`a $m$ \'el\'ements de queues de $E$ est asymptotiquement $1$.
\end{prpsubf}

Nous obtenons aussi la proposition suivante.
\begin{prpsubf}
Soit $E$ la suite des bicarr\'es. ${\fam0 L}^p_E(\T)$ a la \umap/ r\'eelle
seulement si $p=2$ ou $p=4$.
\end{prpsubf}
{\it Preuve.\/ }
$E$ ne satisfait pas la propri\'et\'e de bloc-ind\'ependance $\J{3}$
r\'eelle. En effet, Ramanujan a d\'ecouvert l'\'egalit\'e suivante
pour tout $n$:
$$
(4n^5-5n)^4+(6n^4-3)^4+(4n^4+1)^4=
(4n^5+n)^4+(2n^4-1)^4+3^4.\eqno{\ecks}
$$


\subsection{Impact de la croissance du spectre}
\label{sec:intro:croiss}

Nous d\'emontrons de mani\`ere directe le r\'esultat positif suivant.
\begin{thmsubf}
Soit $E=\{n_k\}\se\Z$ tel que $n_{k+1}/n_k\to\infty$. Alors la suite
des projections associ\'ee \`a $E$ r\'ealise la \umap/ complexe dans
$\SCE$ et $E$ est un ensemble de Sidon de constante asymptotiquement
$1$. Dans l'hypoth\`ese o\`u les rapports $n_{k+1}/n_k$ sont tous
entiers, la r\'eciproque vaut.
\end{thmsubf}
\begin{corsubf}
Alors $E$ est une suite basique inconditionnelle m\'etrique dans tout
espace de Banach homog\`ene $X$ de fonctions sur $\T$. De plus, $X_E$ a la
\umap/ complexe.
\end{corsubf}
{\it Esquisse de preuve.\/ }
Nous prouvons concr\`etement que si $n_{k+1}/n_k\to\infty$, alors quel
que soit $\eps>0$ il existe $l\ge1$ tel que pour toute fonction
$f=\sum a_k\e_{n_k}$
\begin{equation}\label{p6}
\|f\|_\infty\ge(1-\eps)\Bigl(\Bigl\|\sum_{k\le
  l}a_k\e_{n_k}\Bigr\|_\infty+\sum_{k>l}|a_k|\Bigr).
\end{equation}
Cela revient \`a dire que la suite $\{\pi_k\}$ de projections
associ\'ee \`a la base $E$ r\'ealise la $1/(1-\eps)$-\uap/. Pour
obtenir l'in\'egalit\'e $(\ref{p6})$, on utilise une r\'ecurrence
bas\'ee sur l'id\'ee suivante. 

Soit $u\in\T$ tel que $\|\pi_kf\|_\infty=|\pi_kf(u)|$. 
Il existe alors $v\in\T$ tel que
$$ 
|u-v|\le\pi/|n_{k+1}|\quad\mbox{et}\quad
|\pi_kf(u)+a_{k+1}\e_{n_{k+1}}(v)|=\|\pi_kf\|_\infty+|a_{k+1}|.$$
De plus, dans ce cas, 
$$
|\pi_kf(u)-\pi_kf(v)|\le
|u-v|\,\|\pi_kf'\|_\infty\le
\pi|n_k/n_{k+1}|\,\|\pi_kf\|_\infty.
$$ 

En r\'esum\'e, $a_{k+1}\e_{n_{k+1}}$ a le m\^eme argument que $\pi_kf$
tr\`es pr\`es du maximum de $|\pi_kf|$, et $\pi_kf$ varie peu. 

Mais alors 
\begin{eqnarray*}
\|\pi_kf(t)+a_{k+1}\e_{n_{k+1}}\|_\infty&\ge&|\pi_kf(v)+a_{k+1}\e_{n_{k+1}}(v)|\\
&\ge&\|\pi_kf\|_\infty+|a_{k+1}|-\pi|n_k/n_{k+1}|\|\pi_kf\|_\infty\\
&=&(1-\pi|n_k/n_{k+1}|)\|\pi_kf\|_\infty+|a_{k+1}|.
\end{eqnarray*}
On obtient $(\ref{p6})$ en r\'eit\'erant cet argument.\eck

Notre technique donne d'ailleurs l'estimation suivante de la constante
de Sidon des ensembles de Hadamard.
\begin{corsubf}
  Soit $E=\{n_k\}\se\Z$ et $q>\sqrt{\pi^2/2+1}$. Si $|n_{k+1}|\ge
  q|n_k|$, alors la constante de Sidon de $E$ est inf\'erieure ou
  \'egale \`a $1+\pi^2/(2q^2-2-\pi^2)$.
\end{corsubf}
Nous prouvons que cette estimation est optimale au sens o\`u
l'ensemble $E=\{0,1,q\}$, $q\ge2$, a pour constante
d'inconditionnalit\'e r\'eelle dans $\SC{}$ 
$$\bigl(\cos(\pi/(2q)\bigr)^{-1}\ge1+\pi^2/8\,q^{-2}.$$


\section{\sloppy Suites basiques $1$-inconditionnelles d'en\-tr\'ees de ma\-tri\-ce}

Dans ce chapitre, nous cherchons \`a r\'epondre \`a la question
\ref{q:schur}. Nous fournissons une r\'eponse compl\`ete dans le cas
particulier des classes de Schatten $S^p$ avec $p$ entier pair. En
effet, la
technique pr\'esent\'ee dans la section \ref{intro:nmcc} peut \^etre
transf\'er\'ee du cadre des multiplicateurs de Fourier au cadre des
multiplicateurs de Schur. Nous interpr\'etons la condition
combinatoire obtenue \`a l'aide d'objets combinatoires introduits {\it
  ad hoc}.  Notre analyse
aboutit au th\'eor\`eme suivant. 
\begin{thmf}Soit $I\se\N\times\N$.\\
\bm{i} $\{\e_{rc}:(r,c)\in I\}$ est une suite basique $1$-inconditionnelle r\'eelle dans
$S^p$ exactement quand elle est $1$-inconditionnelle complexe et m\^eme
c.b.\ $1$-inconditionnelle complexe.\\
\bm{ii}
$I$ satisfait ces trois propri\'et\'es exactement lorsque $I$ est
``matriciellement $p/2$-ind\'ependant'': 
%pour chaque paire de points
%$q,q'\in I$, il existe au plus un seul chemin sans retour de $q$ \`a
%$q'$ dont les sommets sont dans $I$ et dont les segments suivent le
%treillis $\N\times\N$.
deux points $q,q'\in I$ sont reli\'es par au plus un seul chemin sans
retour sur le r\'eseau $\N\times\N$ dont les au plus $p/2$ sommets
sont dans $I$.
%
\end{thmf}

\section{Constructions al\'eatoires \`a l'int\'erieur de suites
  lacunaires} 

Dans ce chapitre, nous fournissons une preuve nouvelle pour une
construction al\'eatoire d'ensembles lacunaires par Yitzhak Katznelson qui
appartient au folklore de l'analyse harmonique. Nous analysons et
g\'en\'eralisons aussi la construction al\'eatoire d'ensembles
\'equidistribu\'es par Jean Bourgain.

Cela nous permet d'\'etablir le tableau \ref{intro:table} qui classe
les propri\'et\'es de Rosenthal et $\Lambda(p)$ pour tout $p$ selon la
croissance du spectre. Nous montrons alors que la d\'emarche
probabiliste suivie par Katznelson et Bourgain pour construire ces
sous-ensembles de $\Z$ utilise seulement la croissance
``arithm\'etique'' et l'\'equidistribution de la suite des entiers
$\Z$. En fait, ces sous-ensembles peuvent \^etre construits \`a 
l'int\'erieur de suites \'equidistribu\'ees \`a croissance
polyn\^omiale. En particulier, le tableau \ref{intro:table} reste
valable pour l'ensemble des sous-ensembles $E$ d'une suite
polyn\^omiale, ainsi que de la suite des nombres premiers.

Nous fournissons une r\'eponse partielle \`a la question
\ref{intro:random:q2}.
\begin{thmf}
Soit $P$ une suite polyn\^omiale ou la suite des nombres
premiers. Alors il existe une sous-suite $E$ de $P$ qui est
$\Lambda(p)$ pour tout $p$ alors qu'elle ne forme pas un ensemble de
Rosenthal.
\end{thmf}
\vfill
Le chapitre II correspond \`a l'article \cite{ne98} publi\'e dans {\it
  Studia Mathematica} sauf la section \ref{chap1}.\ref{ss:sidon-constant}, 
soumise au {\it Bulletin of the London Mathematical Society}. 
Le chapitre IV a \'et\'e soumis aux {\it
  Annales de l'Institut Fourier}. Le chapitre III fait partie d'une
recherche en cours.



%\end{document}


\baselineskip=15.6pt minus 1.2pt
\chapter{Metric unconditionality and Fourier analysis}\label{chap1}







\section{Introduction}
%
%
%
%
%
We study isometric and almost isometric counterparts to the following
two properties of a separable Banach space $Y$:

\vskipa {\bf (ubs) } $Y$ is the closed span of an unconditional basic
sequence;

\vskipa{\bf (uap) } $Y$ admits an unconditional finite dimensional
expansion of the identity.\vskipb

We focus on the case of translation invariant spaces of functions on
the torus group $\T$, which will provide us with a bunch of natural
examples. Namely, let $E$ be a subset of $\Z$ and $X$ be one of the
spaces $\SLP{}$ $(1\le p<\infty)$ or $\SC{}$. If $\{\e^{{\rm
 i}nt}\}_{n\in E}$ is an unconditional basic sequence (\ubs/ for
short) in $X$, then $E$ is known to satisfy strong conditions of
lacunarity: $E$ must be in Rudin's class \EL{q}, $q=p\vee2$, and a
Sidon set respectively. We raise the following question: what kind of
lacunarity is needed to get the following stronger property:

\vskipa{\bf (umbs) } $E$ is a metric unconditional basic sequence in
$X$: for any $\eps>0$, one may lower its unconditionality constant to
$1+\eps$ by removing a finite set from it.\vskipb

In the case of $\SC{}$, $E$ is a \umbs/ exactly when $E$ is a Sidon set
with constant asymptotically $1$.

In the same way, call $\{T_k\}$ an approximating sequence (\as/ for
short) for $Y$ if the $T_k$'s are finite rank operators that tend
strongly to the identity on $Y$; if such a sequence exists, then $Y$
has the bounded approximation property. Denote by $\Delta
T_k=T_k-T_{k-1}$ the difference sequence of $T_k$. Following
Rosenthal\index{Rosenthal, Haskell Paul}
(see \cite[\S1]{fe80}), we then say that $Y$ has the unconditional
approximation property (\uap/ for short) if it admits an \as/
$\{T_k\}$ such that for some $C$
\begin{equation}\label{intro:uap}
\biggl\|\sum_{k=1}^n
\epsilon_k\Delta T_k
\biggr\|_{\script{L}(Y)} \le C
\qquad\mbox{for all $n$ and scalar $\epsilon_k$ with 
$|\epsilon_k|=1$.} 
\end{equation}
By the uniform boundedness principle, $(\ref{intro:uap})$ means
exactly that $\sum\Delta T_ky$ converges unconditionally for all $y\in
Y$. We now ask the following question: which conditions on $E$ do
yield the corresponding almost isometric (metric for short) property,
first introduced by Casazza and Kalton \cite[\S3]{ck91}~?

\vskipa{\bf (umap) } The span $Y=X_E$ of $E$ in $X$ has the metric
unconditional approximation property: for any $\eps>0$, one may lower
the constant $C$ in $(\ref{intro:uap})$ to $1+\eps$ by choosing an
adequate \as/ $\{T_k\}$.\vskipb

Several kinds of metric, \ie almost isometric properties have been
investigated in the last decade (see \cite{hww93}). There is a common
feature to these notions since 
Kalton's\index{Kalton, Nigel J.}
\cite{ka93}: they can be reconstructed from a 
corresponding interaction between some break and some tail of the
space. We prove that \umap/ is characterized by almost
$1$-unconditionality between a specific break and tail, that we coin
``block unconditionality''.\vskipb

Property \umap/ has been studied by Li \cite{li96} for $X=\SC{}$. He
obtains remarkably large examples of such sets $E$, in particular
Hilbert sets. Thus, the second property seems to be much weaker than
the first (although we do not know whether $\SCE$ has \umap/ for all \umbs/
$E$ in $\SC{}$: for sets of the latter kind, the natural sequence of
projections realizes \uap/ in $\SCE$, but we do not know whether it achieves
\umap/).\vskipb
 
In fact, both problems lead to strong arithmetical conditions on $E$
that are somewhat complementary to the property of quasi-independence
(see \cite[\S3]{pi81}). In order to obtain them, we apply
Forelli's\index{Forelli, Frank} \cite[Prop.\ 2]{fo64} and
Plotkin's\index{Plotkin, A. I.} \cite[Th.\ 1.4]{pl74} techniques 
in the study of isometric operators on ${\fam0 L}^p$: see Theorem
\ref{mub:thm} and Lemma \ref{umap:lem}. This may be done at once for
the projections associated to basic sequences of characters. In the
case of general metric unconditional approximating sequences, however,
we need a more thorough knowledge of their connection with the structure
of $E$: this is the duty of Theorem \ref{sbd:thm}. As in
Forelli's
and Plotkin's results, we obtain that the spaces
$X=\SLP{}$ with $p$ 
an even integer play a special r\^ole. For instance, they are the
only spaces which admit $1$-unconditional basic sequences \DE/ with
more than two elements: see Proposition \ref{mub:isom}.\vskipb

There is another fruitful point of view: we may consider elements of
$E$ as random variables on the probability space $(\T,dm)$. They have
uniform distribution and if they were independent, then our questions
would have trivial answers. In fact, they are strongly dependent: for
any $k,l\in\Z$, Rosenblatt's\index{Rosenblatt, Murray} \cite{ro56}
strong mixing coefficient\index{strong mixing}
%
$$
%
\sup\bigl\{|m[A\cap B]-m[A]m[B]|: A\in\sigma(\e^{\ii kt})\mbox{ and
 }B\in\sigma(\e^{\ii lt}) \bigr\}
%
$$
%
has its maximum value, $1/4$. But lacunarity of $E$ enhances their
independence in several weaker senses (see \cite{be90}). Properties
\umap/ and \umbs/ can be seen as an expression of almost independence
of elements of $E$ in the ``additive sense'', \ie when appearing in
sums. We show their relationship to the notions of pseudo-independence
(see \cite[\S4.2]{mu82}) and almost i.i.d.\ sequences (see
\cite{be87}).\vskipb

The gist of our results is the following: almost isometric properties
for spaces $X_E$ in ``little'' Fourier analysis may be read as a
smallness property of $E$. They rely in an essential way on the
arithmetical structure of $E$ and distinguish between real and complex
properties. In the case of $\SL{2n}{}$, $n$ integer, these arithmetical
conditions are in finite number and turn out to be sufficient, because
the norm of trigonometric polynomials is a polynomial expression in
these spaces. Furthermore, the number of conditions increases with
$n$ in that case. In the remaining cases of $\SLP{}$, $p$ not an even
integer, and $\SC{}$, these arithmetical conditions are infinitely many
and become much more coercive. In particular, if our properties are
satisfied in $\SC{}$, then they are satisfied in all spaces $\SLP{}$,
$1\le p<\infty$. \vskipb
 
We now turn to a detailed discussion of our results: in Section
$\ref{sect:mub}$, we first characterize the sets $E$ and values $p$
such that $E$ is a $1$-unconditional basic sequence in $\SLP{}$ (Prop.\ 
\ref{mub:isom}). Then we show how to treat similarly the almost
isometric case and obtain a range of arithmetical conditions \I{n} on
$E$ (Th.\ \ref{mub:thm}). These conditions turn out to be identical
whether one considers real or complex unconditionality: this is
surprising and in sharp contrast to what happens when $\T$ is replaced
by the Cantor group. They also do not distinguish amongst $\SLP{}$
spaces with $p$ not an even integer and $\SC{}$, but single out $\SLP{}$
with $p$ an even integer: this property does not ``interpolate''. This
is similar to the phenomena of
equimeasurability\index{equimeasurability} (see 
\cite[introduction]{ko91}) and $\script{C}^\infty$-smoothness of
norms\index{smoothness} 
(see \cite[Chapter V]{dgz93}). These facts may also be appreciated
from the point of view of natural renormings\index{renormings} of the
Hilbert space 
$\SLE{2}$.\vskipb

In Section $\ref{sect:ex}$, of purely arithmetical nature, we give
many examples of $1$-uncon\-ditional and metric unconditional basic
sequences through an investigation of property \I{n}. As expected
with lacunary series, number theoretic conditions show up (see
especially Prop.\ \ref{mub:trans}).\vskipb

In Section $\ref{sect:block}$, we first return to the general case of
a separable Banach space $Y$ and show how to connect the metric
unconditional approximation property with a simple property of ``block
unconditionality''. Then a skipped blocking technique invented by
Bourgain\index{Bourgain, Jean} and 
Rosenthal\index{Rosenthal, Haskell Paul} \cite{br80} 
gives a canonical way to construct an \as/ that realizes
\umap/ (Th.\ \ref{block:thm}).\vskipb

In Section \ref{sect:lpmap}, we introduce the $p$-additive approximation
property \lpap/ and its metric counterpart, \lpmap/. It may be
described as simply as \umap/. Then we connect \lpmap/ with the work
of Godefroy, Kalton, Li and Werner \cite{kw95,gkl96} on
subspaces of ${\fam0 L}^p$ which are almost isometric to $\ell_p$.\vskipb

Section \ref{sect:tis} focusses on \uap/ and \umap/ in the case of
translation invariant subspaces $X_E$. The property of block
unconditionality may then be expressed in terms of ``break'' and
``tail'' of $E$: see Theorem \ref{sbd:thm}.\vskipb

In Section $\ref{sect:umap}$, we proceed as in Section
$\ref{sect:mub}$ to obtain a range of arithmetical conditions \J{n}
for \umap/ and metric unconditional \fdd/ (Th.\ \ref{umap:thm} and
Prop.\ \ref{umap:prp:fdd}). These conditions are similar to \I{n},
but are decidedly weaker: see Proposition \ref{arith:csq}$(i)$. This
time, real and complex unconditionality differ; again spaces $\SLP{}$
with even $p$ are singled out.\vskipb

In Section $\ref{sect:arith}$, we continue the arithmetical
investigation begun in Section \ref{sect:ex} with property \J{n} and
obtain many examples for the $1$-un\-con\-di\-tio\-nal and the metric
unconditional approximation property.\vskipb

However, the main result of Section $\ref{sect:positif}$, Theorem
\ref{positif:thm}, shows how a rapid (and optimal) growth condition on
$E$ allows avoiding number theory in any case considered. We therefore
get a new class of examples for \umbs/, in particular Sidon sets of
constant asymptotically $1$, and \umap/. We also prove that
$\SC{\{3^k\}}$ has real \umap/ and that this is due to the oddness of
$3$ (Prop.\ \ref{res:geo}). A sharp estimate of the Sidon constant of
Hadamard sets is obtained as a byproduct (Cor.\ \ref{positif:cor}). We 
compute the Sidon constant of sets with three elements (Th.\ \ref{ubc:THM}).
\vskipb

Section $\ref{sect:comb}$ uses combinatorial tools to give some rough
information about the size of sets $E$ that satisfy our arithmetical
conditions. In particular, we answer a question of Li \cite{li96}: for
$X=\SC{}$ and for $X=\SLP{}$, $p\ne2,4$, the maximal density $d^*$ of
$E$ is zero if $X_E$ has \umap/ (Prop.\ \ref{comb:thm}). For
$X=\SL{4}{}$, our technique falls short of the expected result: we
just know that if $\SL{4}{E\cup\{a\}}$ has \umap/ for every $a\in\Z$,
then $d^*(E)=0$.\vskipb

Section $\ref{sect:proba}$ is an attempt to describe the relationship
between these notions and probabilistic independence. Specifically the
Rademacher and Steinhaus sequences show the way to a connection
between metric unconditionality and the almost i.i.d.\ sequences of
\cite{be87}. We note further that the arithmetical property
\I{\infty} of Section \ref{sect:mub} is equivalent to Murai's
\cite[\S4.2]{mu82} property of pseudo-independence.\vskipb

In Section \ref{sect:resume}, we collect our results on metric
unconditional basic sequences of characters and \umap/ in translation
invariant spaces. We conclude with open questions. \vskipb

 
{\bf Notation and definitions } Sections $\ref{sect:mub}$,
\ref{sect:tis}, \ref{sect:umap} and \ref{sect:positif} will take place
in the following framework. $(\T,dm)$ denotes the compact abelian
group $\{z\in\C:|z|=1\}$ endowed with its Haar measure $dm$; $m[A]$ is
the measure of a subset $A\se\T$. Let $\D=\{-1,1\}$. $\U$ will denote
either the complex ($\U=\T$) or real ($\U=\D$) choice of signs. For a
real function $f$ on $\U$, the oscillation\index{oscillation} of $f$ is
$$
\osc_{\epsilon\in\sU}f(\epsilon)= \sup_{\epsilon\in\sU}f(\epsilon)-
\inf_{\epsilon\in\sU}f(\epsilon).
$$
We shall study homogeneous\index{homogeneous Banach space} Banach
spaces $X$ of functions on $\T$ 
\cite[Chapter I.2]{ka68}, and especially the peculiar behaviour of the
following ones: $\SLP{}$ ($1\le p<\infty$), the space of $p$-integrable
functions with the norm $\|f\|_p=(\int|f|^pdm)^{1/p}$, and $\SC{}$, the
space of continuous functions with the norm
$\|f\|_\infty=\max\{|f(t)|:t\in\T\}$. $\script{M}(\T)$ is the dual of
$\SC{}$ realized as Radon measures on \T.

The dual group $\{\e_n\colon z\mapsto z^n:n\in\Z\}$ of $\T$ is
identified with $\Z$. We write $\mes{B}$ for the cardinal of a set
$B$. For a not necessarily increasing sequence \DEEE/, let \PTE/ be
the space of trigonometric polynomials spanned by [the characters in]
$E$. Let $X_E$ be the translation invariant subspace of those
elements in $X$ whose Fourier transform vanishes off $E$: for all
$f\in X_E$ and $n\notin E$, $\widehat{f}(n)=\int
f(t)\e_{-n}(t)dm(t)=0$. $X_E$ is also the closure of \PTE/ in
homogeneous $X$ \cite[Th.\ 2.12]{ka68}. Denote by $\pi_k:X_E\to X_E$
the orthogonal projection onto $X_{\{\lst n k\}}$. It is given by
$$
\pi_k(f)= \widehat{f}(n_1)\e_{n_1}+\dots+\widehat{f}(n_k)\e_{n_k}.
$$
Then the $\pi_k$ commute. They form an \as/ for $X_E$ if and only
if $E$ is a basic sequence. For a finite or cofinite $F\se E$,
$\pi_F$ is similarly the orthogonal projection of $X_E$ onto $X_F$.

Sections \ref{sect:block} and \ref{sect:lpmap} consider the general
case of a separable Banach space $X$. $B_X$ is the unit ball of $X$
and $\Id$ denotes the identity operator on $X$. For a given sequence
$\{U_k\}$, its difference sequence is $\Delta U_k=U_k-U_{k-1}$ (where
$U_0=0$).

The functional notions of \ubs/, \umbs/
% and the unconditionality
%constants $C_p(E)$ 
are defined in \ref{mub:def}. The functional notions of \as/, \uap/
and \umap/ are defined in \ref{block:def}. Properties \lpap/ and
\lpmap/ are defined in \ref{str:def}. The functional property \UP/ of
block unconditionality is defined in \ref{block:block:def}. The sets
of arithmetical relations $\Zeta^m$ and $\Zeta_n^m$ are defined before
\ref{mub:isom}. The arithmetical properties \I{n} of almost
independence and \J{n} of block independence are defined in
\ref{mub:def:ar} and \ref{arith:def} respectively. The pairing $\XE$
is defined before \ref{mub:lim}.





\section{Metric unconditional basic sequences of characters \umbs/}
\label{sect:mub}







\subsection{Definitions. Isomorphic case}








We start with the definition of metric 
unconditional basic sequences (\umbs/ for short). 
$\U=\T=\{\epsilon\in\C:|\epsilon|=1\}$ (\vs $\U=\D=\{-1,1\}$) 
is the complex (\vs real) choice of signs.
%
\begin{dfnsub}\label{mub:def}
  Let \DE/ and $X$ be a homogeneous Banach space on \T.
  
  \bm{i} \cite{ka48} $E$ is an unconditional basic 
  sequence\index{unconditional basic sequence of characters} 
  \ubs/ in $X$ if there is a constant
  $C$ such that
\begin{equation}\label{umbp:dfn}
\biggl\|\sum_{q\in G}\epsilon_qa_q\e_q\biggr\|_X\le
C\biggl\|\sum_{q\in G}a_q\e_q\biggr\|_X
\end{equation}
for all finite subsets $G\se E$, coefficients $a_q\in\C$ and signs 
$\epsilon_q\in\T$ \(\vs
$\epsilon_q\in\D$\). The infimum of such $C$ is the complex \(\vs
real\) unconditionality\index{unconditionality constant}
%\index{complex unconditionality constant}
%\index{real unconditionality constant}
constant of $E$ in $X$. If $C=1$
works, then $E$ is a complex \(\vs real\)
$1$-\ubs/\index{$1$-unconditional basic sequence of characters} in
$X$.
%
%We write $C^{\fam0 {c}}_p$ \(\vs $C^{\fam0 {r}}_p(E)$\)
%for the infimum of such $C$ with
%complex \(\vs real\) $\epsilon_q$. 

\bm{ii} $E$ is a complex \(\vs real\) metric unconditional basic
sequence\index{metric unconditional basic sequence} \umbs/ in $X$ if
for each $\eps>0$ there is a finite set $F$ such that
%$C^{\fam0 c}_p(E\setminus F)$ 
%\(\vs $C^{\fam0 r}_p(E\setminus F)$\)
the complex \(\vs real\) unconditionality constant of $E\setminus F$
is less than $1+\eps$.
\end{dfnsub}
%
Note that \Z\ itself is an \ubs/ in $\SLP{}$ if and only if $p=2$ by
Khinchin's inequality. The same holds in the
framework of the Cantor group\index{Cantor group} $\D^\infty$ and its dual group of Walsh
functions: their common feature with the $\e_n$ is that their modulus
is everywhere equal to $1$ (see \cite{ke81}). 

The following facts are folklore. 
%
\begin{prpsub}
%
  Let $Y$ be a Banach space.
  
  \bm{i} If $\bigl\|\sum\epsilon_ky_k\bigr\|_Y\le 
            C\bigl\|\sum y_k\bigr\|_Y$ for all
  $\epsilon_k\in\T$ \(\vs $\epsilon_k\in\D$\), then this holds
  automatically for all complex \(\vs real\) $\epsilon_k$ with
  $|\epsilon_k|\le 1$.
  
  \bm{ii} Real\index{real vs.\ complex}\index{complex vs.\ real} and
  complex unconditionality are isomorphically $\pi/2$-equivalent.
%
\end{prpsub}
%
\dem
$(i)$ follows by convexity. $(ii)$ Let us use the fact that the 
complex unconditionality constant of the Rademacher sequence is $\pi/2$ 
\cite{se97}: 
\begin{eqnarray*}
\sup_{\delta_k\in\sT}\Bigl\|\sum\delta_ky_k\Bigr\|_Y
&=&\sup_{y^*\in Y^*}\sup_{\delta_k\in\sT}\sup_{\epsilon_k=\pm1}
\Bigl|\sum\delta_k\langle y^*,y_k\rangle\epsilon_k\Bigr|\\&\le&
\pi/2\sup_{y^*\in Y^*}\sup_{\epsilon_k=\pm1}
\Bigl|\sum\langle y^*,y_k\rangle\epsilon_k\Bigr|=
\pi/2\sup_{\epsilon_k=\pm1}\Bigl\|\sum \epsilon_ky_k\Bigr\|_Y.
\end{eqnarray*}
Taking the Rademacher sequence in $\script{C}(\D^\infty)$, 
we see that $\pi/2$ is optimal.\eck\vskipb
%The constant $2$ above is indeed very bad in many cases. 
%One can improve the constant $2$ above in certain cases: see
%\cite{se97}.

In fact, if $(\ref{umbp:dfn})$ holds, then $E$ is a basis 
of its span in $X$, which is $X_E$
% by Weierstra{\ss}' 
%theorem 
\cite[Th.\ 2.12]{ka68}. 
We have the following relationship between 
the unconditionality 
constants of $E$ in $\SC{}$ and in a homogeneous Banach space $X$ on
$\T$. %$\SLPE$ ($1\le p<\infty$).: 
%
\begin{prpsub}\label{mub:pinf}
%
  Let \DE/ and $X$ be a homogeneous Banach space on $\T$.
  
  \bm{i} The complex \(\vs real\) unconditionality constant of $E$ in
  $X$ is at most the complex \(\vs real\)
unconditionality constant of $E$ in $\SC{}$.%$\SLPE$ for $1\le p<\infty$. 
%in $C^{\fam0 r}_\infty(E)=\sup_{1\le p<\infty}C^{\fam0 r}_p(E)$ 
%and $C^{\fam0 c}_\infty(E)=\sup_{1\le p<\infty}C^{\fam0 c}_p(E)$.

\bm{ii} If $E$ is a \ubs/ \(\vs $1$-\ubs/, \umbs/\) in $\SC{}$, then
$E$ is a
\ubs/ \(\vs $1$-\ubs/, \umbs/\) in $X$.%$\SLP{}$ for all $1\le p<\infty$.
%
\end{prpsub}
%
This follows from the well-known (see {\it e.g.\/} 
\cite{ha87})
%
\begin{lemsub}\label{sbd:multiplier}\index{relative multipliers}
%
  Let \DE/ and $X$ be a homogeneous Banach space on $\T$. Let $T$ be a
  multiplier on $\SCE$. Then $T$ is also a multiplier on $X_E$ and
  $$\|T\|_{\script{L}(X_E)}\le \|T\|_{\script{L}(\script{C}_E)}.$$
%$$
%\|T\|_{\script{L}(\script{C}_E)}=
%\supl_{1\le p<\infty}\|T\|_{\script{L}({\fam0 L}^p_E)}.
%$$
%
\end{lemsub}
%
\dem
The linear functional $f\mapsto Tf(0)$ on $\SCE$ extends to a 
measure $\mu\in\script{M}(\T)$ such that 
$\|\mu\|_\script{M}=\|T\|_{\script{L}(\script{C}_E)}$. Let 
$\check{\mu}(t)=\mu(-t)$. Then $Tf=\check{\mu}*f$ for $f\in\PTE/$ and
$$
\|T\|_{\script{L}(X_E)}
\le\|\check{\mu}\|_\script{M}
=\|T\|_{\script{L}(\script{C}_E)}.\eqno{\ecks}$$
%Furthermore, if $\|Tf\|_p\le C\|f\|_p$ for all $1\le p<\infty$, 
%then $\|Tf\|_\infty\le C\|f\|_\infty$ by passing 
%to the limit.
%\eck\vskipb

\questsub
There is no interpolation 
theorem for such relative multipliers. The forthcoming 
Theorem \ref{mub:thm} shows that there can be no 
metric interpolation\index{relative multipliers!interpolation}. 
Is it possible that one
cannot interpolate 
multipliers at all between $\SLPE$ and $\SLE{q}$~? \vskipb

Note that conversely, \cite{fo82} furnishes the example 
of an \DE/ such that 
the $\pi_k$ are uniformly 
bounded on $\SLE{1}$ but not on $\SCE$.

It is known that $E$ is an \ubs/ 
in $\SC{}$ (\vs in $\SLP{}$) if and only if 
it is a Sidon (\vs \EL{2\vee p}) set. 
To see this, let us recall the relevant definitions.
% 
\begin{dfnsub}\label{mub:sido}
%
  Let \DE/.
  
  \bm{i} \cite{ka57} $E$ is a Sidon\index{Sidon set} set if there is a
  constant $C$ such that
  $$
  \sum_{q\in G}|a_q|\le C\biggl\|\sum_{q\in
    G}a_q\e_q\biggr\|_\infty \mbox{ for all finite $G\se E$ and
    $a_q\in\C$.}
  $$
  The infimum of such $C$ is $E$'s Sidon 
  constant\index{Sidon set!constant}.
  
  \bm{ii} \cite[Def.\ 1.5]{ru60} Let $p>1$. $E$ is a
  \EL{p}\index{Lambda(p) set@\EL{p} set} set if there is a constant
  $C$ such that $\|f\|_p\le C\|f\|_1$ for $f\in\PTE/$.
%
\end{dfnsub}
%
In fact, the Sidon constant of $E$ is the complex unconditionality
constant of $E$ in $\SC{}$. Thus $E$ is a complex \umbs/ 
in $\SC{}$ if and only if tails of $E$ 
have their Sidon 
constant arbitrarily close to $1$. We may also 
say: $E$'s Sidon constant is 
asymptotically $1$.

Furthermore, $E$ is a \EL{2\vee p} set if and only if 
$\SLPE=\SLE{2}$. Therefore \EL{2\vee p} sets are 
\ubs/ in $\SLP{}$. Conversely, if $E$ is an 
\ubs/ in $\SLP{}$, then by 
Khinchin's
%\index{Khinchin's inequality} 
inequality
$$
\biggl\|\sum_{q\in G} a_q\e_q\biggr\|_p^p\approx
\mbox{average}
\biggl\|\sum_{q\in G}\pm a_q\e_q\biggr\|_p^p\approx
\Bigl(\sum_{q\in G}|a_q|^2\Bigr)^{p/2}=
\biggl\|\sum_{q\in G} a_q\e_q\biggr\|_2^p
$$
for all finite $G\se E$ (see 
\cite[proof of Th.\ 3.1]{ru60}). This shows also that the \EL{p} 
set constant and the unconditionality constant in $\SLP{}$ are connected
{\it via}\/ the constants in Khinchin's inequality; whereas Sidon sets
have their unconditionality constant in $\SLP{}$ uniformly bounded, the
\EL{p} set constant\index{Lambda(p) set@\EL{p} set!constant} 
of infinite sets grows at least like $\sqrt{p}$ 
\cite[Th.\ 3.4]{ru60}.





\subsection{Isometric case: $1$-unconditional basic sequences of
 characters}\label{ss:isom}






The corresponding isometric question: when is $E$ a complex
$1$-\ubs/~? admits a rather easy answer. To 
this end, introduce the following notation for arithmetical relations\index{arithmetical relation}:
let 
$\Alpha_n=\bigl\{\alpha=\{\alpha_p\}_{p\ge1}:
\alpha_p\in\N\ \&\ 
\alpha_1+\alpha_2+\dots=n\bigr\}$. If 
$\alpha\in \Alpha_n$, all but a finite number of the 
$\alpha_p$ vanish and the multinomial number 
$$\ds\bi n\alpha=\frac{n!}{\alpha_1!\alpha_2!\dots}$$
is well defined. Let 
$\Alpha_n^m=\{\alpha\in \Alpha_n: \alpha_p=0\mbox{ for }p>m\}$.
Note that $\Alpha_n^m$ is finite.
We call $E$ $n$-independent\index{independent set of integers} if 
every integer admits at most one representation as the sum of $n$
elements of $E$, up to a permutation. In terms of arithmetical
relations, this yields
$$
\sum\alpha_ip_i=\sum\beta_ip_i\imp\alpha=\beta\mbox{ for
 $\alpha,\beta\in \Alpha_n^m$ and distinct $\lst pm\in E$}.
$$
This notion is studied in \cite{dr75} where it is called
birelation\index{birelation}. In Rudin's\index{Rudin, Walter}
\cite[\S$1.6(b)$]{ru60} 
notation, the number $r_n(E;k)$ of
representations of $k\in\Z$ as a sum of $n$ elements of $E$ is at most
$n!$ for all $k$ if $E$ is $n$-independent (the converse if false).
This may also be expressed in the framework of arithmetical 
relations 
$$\Zeta^m=\{\zeta\in{\Z^*}^m:\sm\zeta m=0\}\quad\&\quad\Zeta_n^m=\{\zeta\in\Zeta^m:|\zeta_1|+\dots+|\zeta_m|\le 2n\}.$$
Note that $\Zeta_n^m$ is finite, and void if $m>2n$. 
Then $E$ is $n$-independent if and only if 
$$
\sum\zeta_ip_i\ne0\quad\hbox{for all }\zeta\in\Zeta_n^m
\hbox{ and distinct }\lst pm\in E.
$$ 
We shall prefer 
to treat arithmetical relations in terms of $\Zeta_n^m$ rather 
than $\Alpha_n^m$. 
%
\begin{prpsub}\label{mub:isom}
\index{$1$-unconditional basic sequence of characters!in spaces $\SLP{}$, $p$ even} 
\index{$1$-unconditional basic sequence of characters!in $\SC{}$ and $\SLP{}$, $p\notin2\N$} 
  Let \DE/.
 
  \bm{i} $E$ is a complex $1$-\ubs/ in $\SLP{}$, $p$ not an even
  integer, or in $\SC{}$, if and only if $E$ has at most two elements.
 
  \bm{ii} If $p$ is an even integer, then $E$ is a complex $1$-\ubs/
  in $\SLP{}$ if and only if $E$ is $p/2$-independent. There is a
  constant $C_p>1$ depending only on $p$, such that either $E$ is a
  complex $1$-\ubs/ in $\SLP{}$ or the complex unconditionality
  constant of $E$ in $\SLP{}$ is at least $C_p$.
%
\end{prpsub}
%
\dem
%
$(i)$ By Proposition \ref{mub:pinf}$(ii)$, if $E$ is not a complex
$1$-\ubs/ in some $\SLP{}$, then neither in $\SC{}$. 
Let $p$ be not an even integer. We may suppose 
$0\in E$; let $\{0,k,l\}\se E$. If we had
$\|1+\mu a\e_k+\nu b\e_l\|_p=\|1+a\e_k+b\e_l\|_p$ 
for all $\mu,\nu\in\T$, then 
\begin{eqnarray*}
\int|1+a\e_{k}+b\e_{l}|^pdm&=&
\int|1+\mu a\e_{k}+\nu b\e_{l}|^pdm(\mu)dm(\nu)dm\\
&=&
\int|1+\mu a+\nu b|^pdm(\mu)dm(\nu).
\end{eqnarray*}
Denoting by $\theta_i\colon(\epsilon_1,\epsilon_2)\mapsto\epsilon_i$ 
the projections of $\T^2$ onto $\T$, this would mean that
$\|1+a\e_{k}+b\e_{l}\|_p
=\|1+a\theta_1+b\theta_2\|
_{{\fam0 L}^p(\sT^2)}$ 
for all $a,b\in\C$. By 
\cite[Th.\ I]{ru76}, 
$(\e_{k},\e_{l})$ and $(\theta_1,\theta_2)$ 
would have the
same distribution. This is false, since 
$\theta_1$ and $\theta_2$ are independent random variables 
while $\e_{k}$ and $\e_{l}$ are not. 

$(ii)$
Let $\lst qm\in E$ be distinct and 
$\lst\epsilon m\in\T$. 
By the multinomial formula for the power $p/2$ and 
Bessel--Parseval's formula, we get
\begin{eqnarray}
\nonumber
\lefteqn{\Biggl\|\sum_{i=1}^m\epsilon_ia_i\e_{q_i}\Biggr\|_p^p
=\int\Biggl|\sum_{\alpha\in \Alpha_{p/2}^m}
\bip{p/2}\alpha\prod_{i=1}^m(\epsilon_ia_i)^{\alpha_i}
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}\Biggr|^2dm}\\
\nonumber&=&\sum_{A\in\script{R}_q}\Biggl|
\sum_{\alpha\in A}
\bip{p/2}\alpha
\prod_{i=1}^m(\epsilon_ia_i)^{\alpha_i}\Biggr|^2\\
\nonumber&=&\sum_{\alpha\in \Alpha_{p/2}^m}
{\bip{p/2}\alpha}^2\prod_{i=1}^m|a_i|^{2\alpha_i}
\label{cp>1}+
%\sum_{A\in\script{R}_q}\sum_{\alpha\ne\beta\in A}
\sum_{\scriptstyle\alpha\ne\beta\in \Alpha_{p/2}^m\atop
\scriptstyle\alpha\sim\beta}
\bip{p/2}\alpha\bip{p/2}\beta
\prod_{i=1}^m\epsilon_i^{\alpha_i-\beta_i}
a_i^{\alpha_i}\overline{a_i}^{\beta_i},
\end{eqnarray}
where $\script{R}_q$ is the partition of $\Alpha_{p/2}^m$ 
induced by the equivalence relation 
$\alpha\sim\beta\Leftrightarrow
\sum\alpha_iq_i=\sum\beta_iq_i$. If $E$ is 
$p/2$-independent, the second sum in $(\ref{cp>1})$ 
is void and $E$ is a 
$1$-\ubs/.

Furthermore, suppose $E$ is not $p/2$-independent and 
let $\lst qm\in E$ be a minimal set of distinct elements of $E$ 
such that there are $\alpha,\beta\in \Alpha_{p/2}^m$ with 
$\alpha\sim\beta$. Then $m\le p$. Take $a_i=1$ in 
the former computation: then the clearly nonzero 
oscillation of 
$(\ref{cp>1})$ for $\lst\epsilon m\in\T$ does only 
depend on $\script{R}_q$ and thus is finitely valued. 
This yields $C_p$.
%
\eck\vskipb

\exasub\index{unconditionality constant!in $\SL{4}{}$}
Let us treat explicitly the case $p=4$. If $E$ is not $2$-independent,
then one of the two following arithmetic relations occurs on $E$:
$$
2q_1=q_2+q_3\quad\hbox{or}\quad q_1+q_2=q_3+q_4.
$$
In the first case, we may assume $q_2<q_1<q_3$ and thus
$$
2q_2<q_1+q_2<2q_1=q_2+q_3<q_1+q_3<2q_3.
$$
Let $\varrho>0$. Then
$$
\int{|\e_{q_1}+\varrho\e_{q_2}+\epsilon\varrho\e_{q_3}|}^4dm=
1+6\varrho^4+4\varrho^2(2+\Re\epsilon).
$$
Taking $\epsilon=-1$ and $\epsilon=1$, $\varrho=6^{-1/4}$, we see that
$E$'s real 
unconditionality constant is at least the fourth root of $2\sqrt{6}-3$. In fact, $E$'s real and
complex unconditionality constants coincide with this value.

In the second case, we may assume $q_1<q_3<q_4<q_2$ and thus
$$
2q_1<q_1+q_3<q_1+q_4,2q_3<q_1+q_2=
q_3+q_4<q_2+q_3,2q_4<q_2+q_4<2q_2.
$$
We may further assume $q_1+q_4\ne2q_3$ and $q_2+q_3\ne2q_4$: 
otherwise the first case occurs. Then
$$
\int{|\e_{q_1}+\e_{q_2}+\e_{q_3}+\epsilon\e_{q_4}|}^4dm=28+8\Re\epsilon.
$$
Thus $E$'s real
unconditionality constant must be at least $(9/5)^{1/4}$. In fact,
$E$'s real and 
complex unconditionality constants coincide with this value.

From these two cases we conclude that $C_2=(9/5)^{1/4}\approx1.16$ is the optimal
choice 
for the constant in Proposition \ref{mub:isom}$(ii)$.\vskipb

\remsub
We shall compute explicitly the Sidon constant of sets with three
elements and show that it is equal to the real unconditionality
constant in that case. This provides an alternative proof and a
generalization of Prop.\ \ref{mub:isom} $(i)$ for $\SC{}$.\vskipb

\remsub\edef\remre{\thethmsub}
In fact the conclusion in $(ii)$ holds also 
if we assume that $E$ is just a
real\index{real vs.\ complex}\index{complex vs.\ real} 
$1$-\ubs/. If we have some arithmetical 
relation $\alpha\sim\beta$, we may 
assume that $\alpha_i-\beta_i$ is 
odd for one $i$ at least. 
Indeed, we may simplify all 
$\alpha_i-\beta_i$ by their 
greatest common divisor and this 
yields another arithmetical 
relation $\sum(\alpha'_i-\beta'_i)q_i=0$. 
But then the 
oscillation of 
$(\ref{cp>1})$ is again clearly nonzero for 
$\lst\epsilon m\in\D$. \vskipb

\remsub
We shall see in Remark \ref{umbs:remRE} that $(i)$ also holds 
in the real\index{real vs.\ complex}\index{complex vs.\ real} setting. This is a property of $\T$ 
and fails for the Cantor\index{Cantor group} group $\D^\infty$: 
the Rademacher
sequence forms a real $1$-\ubs/ 
in $\script{C}(\D^\infty)$ but is clearly not complex 
$1$-unconditional in any space ${\fam0 L}^p(\D^\infty)$, $p\ne2$: 
see Section \ref{sect:proba} and \cite{se97}. \vskipb

\questsub
There are nevertheless subspaces of $\SLP{}$, $p$ not an even integer,
and $\SC{}$ with $1$-unconditional
bases\index{$1$-unconditional basic sequence of characters!in $\SC{}$ and $\SLP{}$, $p\notin2\N$}, 
like sequences of functions with disjoint support. What
about spaces $\SLPE$ and $\SCE$, in particular when $E$ is finite ? Are there
$1$-unconditional bases that do not consist of characters ?\vskipb

\remsub
For each even integer $p\ge4$, there are $p/2$-independent sets that
are not 
\EL{p+\eps} for any $\eps>0$: such maximal \EL{p} sets are
constructed in \cite{ru60}.\vskipb 








\subsection{Almost isometric case. A computation}
\label{ss:almost}







As $1$-\ubs/ are thus a quite exceptional 
phenomenon and distinguish so harshly 
between even integers and all other reals, one may 
wonder what kind of behaviour its almost isometric counterpart will 
bring about. In the proof 
of Proposition \ref{mub:isom}$(i)$, we used the fact that 
the $\e_n$, seen as random variables, are dependent: 
the ${\fam0 L}^p$ norm for even integer $p$ is just somewhat 
blind to this because it keeps the interaction of 
the random variables down to a finite number of 
arithmetical relations\index{arithmetical relation}. 
The contrast with the other 
${\fam0 L}^p$ norms becomes clear when we try to compute 
explicitly an expression of type 
$\bigl\|\sum\epsilon_qa_q\e_q\bigr\|_p$ for any $p\in[1,\infty\mathclose[$. 
This sort of seemingly brutal computation has been 
applied successfully in \cite[Prop.\ 2]{fo64} and 
\cite[Th.\ 1.4]{pl74} to 
study isometric operators\index{isometries on ${\fam0 L}^p$} on
${\fam0 L}^p$, $p$ not 
an even integer.

We now undertake this tedious computation as preparatory 
work for Theorem \ref{mub:thm}, 
Lemma \ref{umap:lem} and Proposition \ref{umap:prp:fdd}. 
Let us fix some more notation:
for $x\in\R$ and $\alpha\in \Alpha_n$, 
put 
$$\bi x\alpha=\bi xn\bi n\alpha.$$
This generalized 
multinomial number is nonzero if and only if $x\ge n$ or 
$x\notin\N$.
%
\begin{ldcsub}\label{mub:calcul}
%
  Let $\U=\T$ or $\U=\D$ in the complex and real case respectively.
  Let $1\le p<\infty$ and $m\ge1$. Put
  $$
  \varphi_q(\epsilon,z,t)=\biggl|1+\sum_{i=1}^m
  \epsilon_iz_i\e_{q_i}(t)\biggr|^p\quad,\quad \Phi_q(\epsilon,z)=
  \int\varphi_q(\epsilon,z,t)\,dm(t)
  $$
  for $q=(\lst qm)\in\Z^m$, $\epsilon=(\lst\epsilon m)\in\U^m$ and
  $z=(\lst zm)\in D^m$, where $D$ is the disc $\{|w|\le\varrho\}\se\C$
  for some $0<\varrho<1/m$.  
%Let $\script{R}_q$ be the partition of
%  $\N^m$ induced by 
Define the equivalence relation $\alpha\sim\beta\Leftrightarrow
\sum\alpha_iq_i=\sum\beta_iq_i$. Then
%
\begin{eqnarray}
%
\nonumber\Phi_q(\epsilon,z)&=&
\sum_{\alpha\in\sN^m}
{\bip{p/2}\alpha}^2\prod|z_i|^{2\alpha_i}
\label{mub:phi}+
\sum_{\scriptstyle\alpha\ne\beta\in\sN^m\atop\scriptstyle\alpha\sim\beta}
%\sum_{A\in\script{R}_q}\sum_{\alpha\ne\beta\in A}
\bip{p/2}\alpha\bip{p/2}\beta
\prod z_i^{\alpha_i}
\overline{z_i}^{\beta_i}
\epsilon_i^{\alpha_i-\beta_i}.
%
\end{eqnarray}
%
Furthermore, $\{\Phi_q:q\in\Z^m\}$ is a relatively compact subset of
$\script{C}^\infty(\U^m\times D^m)$.
%
\end{ldcsub}
%
\dem 
%
The function $\Phi_q$ is infinitely 
differentiable on the compact set 
$\U^m\times D^m$. Furthermore the family 
$\{\Phi_q:\lst qm\in\Z\}$ 
is bounded in 
$\script{C}^\infty(\U^m\times D^m)$ and henceforth 
relatively compact by Montel's theorem.
Let us compute $\varphi_q$. 
By the expansion of the function $(1+w)^{p/2}$, 
analytic on the unit disc, and the multinomial formula, we have
\begin{eqnarray*}
\varphi_q(\epsilon,z)
&=&\biggl|\sum_{a\ge0}\bip{p/2}a
\biggl(\sum_{i=1}^m\epsilon_iz_i\e_{q_i}
\biggr)^a\biggr|^2\\
&=&\biggl|
\sum_{a\ge0}\bip{p/2}a
\sum_{\alpha\in \Alpha_a^m}\bi a\alpha
\prod(\epsilon_iz_i)^{\alpha_i}
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}
\biggr|^2\\
&=&\biggl|\sum_{\alpha\in\sN^m}
\bip{p/2}\alpha
\prod(\epsilon_iz_i)^{\alpha_i}
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}\biggr|^2.
\end{eqnarray*}
Let $\script{R}_q$ be the partition of $\N^m$ induced by $\sim$. Then,
by Bessel--Parseval's formula 
$$
\Phi_q(\epsilon,z)=\sum_{A\in\script{R}_q}
\biggl|\sum_{\alpha\in A}
\bip{p/2}\alpha\prod(\epsilon_iz_i)^{\alpha_i}
\biggr|^2
$$
and this gives $(\ref{mub:phi})$ by expanding the modulus.
\eck\vskipb

\remsub\edef\remar{\thethmsub}
If $m\ge2$, this expansion has a finite number of terms if and only if $p$ 
is an even integer: then and only then 
$\bip{p/2}\alpha=0$ for $\sum\alpha_i>p/2$, whereas 
$\script{R}_q$ contains clearly some class with 
two elements and thus an infinity thereof. For example,
we have the following arithmetical 
relation\index{arithmetical relation} on $q_1,q_2$ or 
$q_1,q_2,0$ respectively: 
$$
\overbrace{q_1+\dots+q_1}^{|q_2|}=
\overbrace{q_2+\dots+q_2}^{|q_1|}\qquad\mbox{if }\sgn q_1=\sgn q_2;
$$
$$
\overbrace{q_1+\dots+q_1}^{|q_2|}+\overbrace{q_2+\dots+q_2}^{|q_1|}
=0\qquad\mbox{if not.}
$$

\remsub\label{umbs:remRE}
This shows that Proposition \ref{mub:isom}$(i)$ 
holds also in the real setting: we may suppose that 
$0\in E$; take $m=2$ and choose
$q_1,q_2\in E$. One of the two relations 
in Remark \remar\ yields an arithmetical relation on $E$ 
with at least one odd coefficient, 
as done in Remark \remre. But then
$(\ref{mub:phi})$ contains terms nonconstant in 
$\epsilon_1\in\D$ or in $\epsilon_2\in\D$ and thus $E$ cannot be a real
$1$-unconditional basic sequence in $\SLP{}$.\vskipb

We return to our computation. 
%
\begin{ldcsub}\label{mub:culcul}
%
  Let $r=(\lstp r0m)\in E^{m+1}$ and put $q_i=r_i-r_0$ $(1\le i\le
  m)$.  Define
%
\begin{equation}\label{arith:Theta}
%
\Theta_r(\epsilon,z)=\int
\biggl|\e_{r_0}+\sum_{i=1}^m\epsilon_iz_i\e_{r_i}\biggr|^p=
\Phi_q(\epsilon,z)
%
\end{equation}
%
Let $\lstp\zeta0m\in\Z^*$ and
\begin{equation}\label{mub:gammadelta}
(\gamma_i,\delta_i)=
(-\zeta_i\vee0,\zeta_i\vee0)\qquad(1\le i\le m).
\end{equation} 
If the arithmetical relation
%
\begin{equation}\label{mub:ArithRel}
%
\zeta_0r_0+\dots+\zeta_mr_m=0
\qquad\mbox{while}\qquad\smp\zeta0m=0
%
\end{equation}
%
holds, then the coefficient of $\prod z_i^{\gamma_i}
\overline{z_i}^{\delta_i} \epsilon_i^{\gamma_i-\delta_i}$ in
$(\ref{mub:phi})$ is $\bip{p/2}\gamma\bip{p/2}\delta$ and thus
independent of $r$. If $\sum|\zeta_i|\le p$ or $p$ is not an even
integer, this coefficient is nonzero.
%
\end{ldcsub}
%
\dem We have 
$\delta_i-\gamma_i=\zeta_i$, 
$\sum\gamma_i-\sum\delta_i=\zeta_0$ and 
$\sum\gamma_i+\sum\delta_i=|\zeta_1|+\dots+|\zeta_m|$, 
so that $\sum\gamma_i\vee\sum\delta_i=\frac12\sum|\zeta_i|$. 
Moreover $\sum(\delta_i-\gamma_i)q_i=\sum\zeta_ir_i=0$, 
so that $\gamma\sim\delta$.
%
\eck\vskipb







\subsection{Almost independent sets of integers. Main theorem}







The Computational lemmas suggest the following definition. 
%
\begin{dfnsub}\label{mub:def:ar}
%
  Let \DE/.
  
  \bm{i}
%
  $E$ enjoys the property \I{n} of almost
  $n$-independence\index{almost independence} provided there is a
  finite subset $G\se E$ such that $E\setminus G$ is $n$-independent,
  \ie $\zeta_1r_1+\dots+\zeta_mr_m\ne0$ for all $\zeta\in\Zeta_n^m$
  and $\lst rm\in E\setminus G$.
  
  \bm{ii}
%
  $E$ enjoys exactly \I{n} if furthermore it fails \I{n+1}.
  
  \bm{iii}
%
  $E$ enjoys \I{\infty} if it enjoys \I{n} for all $n$, \ie for any
  $\zeta\in\Zeta^m$ there is a finite set $G$ such that
  $\zeta_1r_1+\dots+\zeta_mr_m\ne0$ for $\lst rm\in E\setminus G$.
%
\end{dfnsub}
%
Note that property \I{1} is void and that $\I{n+1}\imp\I{n}$. 
This property is also stable under unions with a finite set. 
The preceding computations yield 
%
\begin{thmsub}\label{mub:thm}
  \index{metric unconditional basic sequence} Let \DEE/ and $1\le
  p<\infty$.
  
  \bm{i} Suppose $p$ is an even integer. Then $E$ is a real, and at
  the same times complex, \umbs/ in $\SLP{}$ if and only if $E$ enjoys
  \I{p/2}. If \I{p/2} holds, there is in fact a finite $G\se E$ such
  that $E\setminus G$ is a $1$-\ubs/ in $\SLP{}$.
  
  \bm{ii} If $p$ is not an even integer and $E$ is a real or complex
  \umbs/ in $\SLP{}$, then $E$ enjoys \I{\infty}.
\end{thmsub}
\dem
Sufficiency in $(i)$ follows directly from 
Proposition \ref{mub:isom}: 
if $E\setminus G$ is $p/2$-independent, then 
$E\setminus G$ is a real and complex $1$-\ubs/.

Let us prove the necessity of the arithmetical property.
We keep the notation of Computational lemmas 
\ref{mub:calcul} and \ref{mub:culcul}. 
Assume $E$ fails \I{n} and let 
$\lstp\zeta0m\in\Z^*$ with $\sum\zeta_i=0$ 
and $\sum|\zeta_i|\le2n$ such that for each $l\ge1$ there 
are distinct $\lstp{r^l}0m\in E\setminus\{\lst nl\}$ with 
$\zeta_0r_0^l+\dots+\zeta_mr_m^l=0$. One may 
furthermore assume that at least one of the $\zeta_i$ 
is not even. 

Assume $E$ is a \umbs/ in $\SLP{}$. Then the oscillation of 
$\Theta_r$ in $(\ref{arith:Theta})$ satisfies 
\begin{equation}\label{mub:ApplDef}
\osc_{\epsilon\in\sU^m}
\Theta_{r^l}(\epsilon,z)\tol_{l\to\infty}0
\end{equation}
for each $z\in D^m$. We may assume that 
the sequence of functions $\Theta_{r^l}$ converges 
in $\script{C}^\infty(\U^m\times D^m)$ to a function 
$\Theta$. Then by $(\ref{mub:ApplDef})$,
$\Theta(\epsilon,z)$ is constant in $\epsilon$ 
for each $z\in D^m$: in particular, its coefficient of 
$\prod z_i^{\gamma_i}
\overline{z_i}^{\delta_i}
\epsilon_i^{\gamma_i-\delta_i}$ is zero. 
(Note that at least one of the $\gamma_i-\delta_i$ is not 
even). 
This is impossible by Computational lemma \ref{mub:culcul} 
if $p$ is either not an even integer or if $p\ge2n$.
%
\eck
%
\begin{corsub}\label{arith:cor}
  \index{Sidon set!with constant asymptotically $1$} Let \DE/. If $E$
  is a \umbs/ in $\SC{}$, that is $E$'s Sidon constant is
  asymptotically $1$, then $E$ enjoys \I{\infty}. The converse does
  not hold.
%
\end{corsub}
%
\dem
%
Necessity follows from Theorem \ref{mub:thm} and 
Proposition \ref{mub:pinf}$(ii)$. 
There is a counterexample to the converse in 
\cite[Th.\ 4.11]{ru60}: Rudin constructs a set $E$ that 
enjoys \I{\infty} while $E$ is not even a Sidon\index{Sidon set} set.
%
\eck\vskipb

%\remsub
%We shall use the same method to study \umap/ in spaces $X_E$. Note that
%this method might be applied to other similar properties., like the
%following. Let \DEE/. For every $\eps>0$ there is a $k$ such that for
%every polynomial $f=\sum a_j\e_{n_j}$ and every choice of signs
%$\epsilon_j$ $(j>k)$,
%$\|\sum_{j\le k}a_je_{n_j}+\sum_{j>k}\epsilon_ja_je_{n_j}\|_X\le\|f\|_X$.
%This property has the advantage of 
%taking into account the ``break'' of $f$: because of this, it is a
%stronger property than complex \umap/, and this, as we shall see, is
%not the case for \umbs/.

For $p$ an even integer, Sections \ref{sect:ex} 
and \ref{sect:comb} will provide various examples of 
\umbs/ in $\SLP{}$. 
Proposition \ref{comb:grow} gives a 
general growth condition on $E$ under which it is an \umbs/. 

As we do not know any partial converse to 
Theorem \ref{mub:thm}$(ii)$ and Corollary \ref{arith:cor}, the sole 
known examples 
of \umbs/ in $\SLP{}$, $p$ not an 
even integer, and $\SC{}$ are those given by 
Theorem \ref{positif:thm}. This theorem will 
therefore provide us with Sidon sets of constant 
asymptotically $1$. Note, however, that Li 
\cite[Th.\ 4]{li96} already constructed 
implicitly such a Sidon set by using 
Kronecker's theorem. 









\section{Examples of metric unconditional basic 
s\-e\-q\-u\-e\-n\-c\-e\-s}\label{sect:ex}






After a general study of the arithmetical property 
of almost independence \I{n}, we shall investigate 
three classes of subsets of $\Z$: integer 
geometric sequences, more generally integer parts of 
real geometric sequences, and polynomial sequences.







\subsection{General considerations}
\label{ss:mubs:gen}







The quantity
%
\begin{eqnarray*}
%
\XE&=&\sup_{G\se E\hbox{\scriptsize\ finite}}\,
\inf\bigl\{|\zeta_1p_1+\dots+\zeta_mp_m|:
\lst p m\in E\setminus G\hbox{ distinct}\bigr\}\\
&=&\lim_{l\to\infty}\inf\bigl\{|\zeta_1p_1+\dots+\zeta_mp_m|:
\lst p m\in\{\lstf nl\}\hbox{ distinct}\bigr\},
%
\end{eqnarray*}
%
where $\{n_k\}=E$, plays a key r\^ole. We have
%
\begin{prpsub}\label{mub:lim}
%
  Let \DEE/.
  
  \bm{i} $E$ enjoys \I{n} if and only if $\XE\ne0$ for all
  $\zeta\in\Zeta_n^m$.  If $\XE<\infty$ for some $\lst\zeta m\in\Z^*$,
  then $E$ fails \I{|\zeta_1|+\dots+|\zeta_m|}. Thus $E$ enjoys
  \I{\infty} if and only if $\XE=\infty$ for all $\lst\zeta m\in\Z^*$.
  
  \bm{ii} Suppose $E$ is an increasing sequence. If $E$ enjoys \I{2},
  then the pace $n_{k+1}-n_k$ of $E$ tends to infinity.
  
  \bm{iii} Suppose $jF+s,kF+t\se E$ for an infinite $F$, $j\ne
  k\in\Z^*$ and $s,t\in\Z$. Then $E$ fails \I{|j|+|k|}.
%
  
  \bm{iv} Let $E'=\{n_k+m_k\}$ with $\{m_k\}$ bounded. Then
  $\XE=\infty$ if and only if $\X{E'}=\infty$. Thus \I{\infty} is
  stable under bounded perturbations of $E$.

\end{prpsub}
%
\dem
%
$(i)$
%
Suppose $\XE<\infty$. Then there is an $h\in\Z$ such that there are 
sequences $\lst{p^l}m\in\{n_k\}_{k\ge l}$ with 
$\sum\zeta_ip_i^l=h$ and $\{\lst{p^{l+1}}m\}$ is disjoint from
$\{\lst{p^l}m\}$ for all $l\ge1$. As
$\sum\zeta_ip_i^l-\sum\zeta_ip_i^{l+1}=0$ for $l\ge1$,
$E$ fails \I{|\zeta_1|+\dots+|\zeta_m|}.

$(ii)$
Indeed, $\langle (1,-1),E\rangle=\infty$.

$(iii)$
Put $\zeta=(j,-k)$. Then $\XE<\infty$.
%
\eck\vskipb






\subsection{Geometric sequences}\index{geometric sequences}









Let $G=\{j^k\}_{k\ge0}$ with
$j\in\Z\setminus\{-1,0,1\}$.
Then $G,jG\se G$: so $G$
fails \I{|j|+1}. 
In order to check \I{|j|} for $G$, 
let us study more carefully the following Diophantine equation:
%
\begin{equation}\label{arith:dioph}
%
%\zeta_1j^{k_1}+\dots+\zeta_mj^{k_m}=0\quad
%\hbox{with}\quad$|\zeta_1|+\dots+|\zeta_m|\le2|j|$\ \&\
%$k_1<\dots<k_m$.}
%
\sum_{i=1}^m\zeta_ij^{k_i}=0\quad\hbox{with}\quad
\zeta\in\N^*\times{\Z^*}^{m-1}\
\&\ \sum_{i=1}^m|\zeta_i|\le2|j|\ \&\ k_1<\dots<k_m.
%
\end{equation}
%
Suppose $(\ref{arith:dioph})$ holds.
Then necessarily $m\ge2$ and
$\zeta_1+\sum_{i=2}^m\zeta_ij^{k_i-k_1}=0$. Hence
$j\mid\zeta_1$ and $\zeta_1\ge|j|$. As $\zeta_1<2|j|$, $\zeta_1=|j|$. Then 
$\sgn j+\sum_{i=2}^m\zeta_ij^{k_i-k_1-1}=0$. Hence
$k_2=k_1+1$ and $j\mid\sgn j+\zeta_2$. As $|\zeta_2|\le|j|$,
$\zeta_2\in\{-\sgn j,j-\sgn j\}$. If $\zeta_2=j-\sgn j$, then
$m=3$, $k_3=k_1+2$ and $\zeta_3=-1$. If $\zeta_2=-\sgn j$,
then $m=2$: otherwise, $j\mid\zeta_3$ as before and
$|\zeta_1|+|\zeta_2|+|\zeta_3|>2|j|$. Thus
$(\ref{arith:dioph})$ has exactly two solutions:
%
\begin{equation}\label{arith:sol}
%
|j|\cdot j^k+(-\sgn j)\cdot j^{k+1}=0\ \&\ 
|j|\cdot j^k+(j-\sgn j)\cdot j^{k+1}+
(-1)\cdot j^{k+2}=0.
%
\end{equation}
%
If $j$ is positive, this shows that $G$ enjoys \I{j}: 
both solutions yield $\sum\zeta_i\ne0$. If $j$
is negative, $G$ enjoys \I{|j|-1}, but the second
solution of $(\ref{arith:dioph})$ shows that $G$
fails \I{|j|}.\vskipb







\subsection{Algebraic and transcendental numbers}
\index{transcendental numbers}








An interesting feature of property \I{\infty}
is that it distinguishes between algebraic and
transcendental numbers. A similar fact has
already been noticed by Murai\index{Murai, Takafumi}
\cite[Prop.\ 26, Cor.\ 28]{mu82}.
%
\begin{prpsub}\label{mub:trans}
%
  Let \DEE/.
  
  \bm{i} If $n_{k+1}/n_k\to\sigma$ where $\sigma>1$ is transcendental,
  then $\XE=\infty$ for any $\lst\zeta m\in\Z^*$. Thus $E$ enjoys
  \I{\infty}.
  
  \bm{ii} Write $[x]$ for the integer part of a real $x$.  Let
  $n_k=[\sigma^k]$ with $\sigma>1$ algebraic. Let
  $P(x)=\zeta_0+\dots+\zeta_dx^d$ be the corresponding polynomial of
  minimal degree. Then $\XE<\infty$ and $E$ fails
  \I{|\zeta_0|+\dots+|\zeta_d|}.
%
\end{prpsub}
%
Note that part $(ii)$ is very restrictive on the speed of
convergence of $n_{k+1}/n_k$ to $\sigma$: even if we take into account
Proposition \ref{mub:lim}$(iv)$, it requires that
$$|n_{k+1}/n_k-\sigma|\preccurlyeq\sigma^{-k}.$$

\dem
$(i)$
Suppose on the contrary that we have $\zeta$ and
sequences $p_1^l<\dots<p_m^l$ in $E$ that tend to
infinity such that $\zeta_1p_1^l+\dots+\zeta_mp_m^l=0$.
As the sequences $\{p_i^l/p_m^l\}_l$ $(1\le i\le m)$
are bounded, we
may assume they are converging ---~and by hypothesis,
they converge either to $0$, say for $i<j$, or to
$\sigma^{-d_i}$ for $d_i\in\N$ and $i\ge j$. But then
$\zeta_j\sigma^{-d_j}+\dots+\zeta_m\sigma^{-d_m}=0$
and $\sigma$ is algebraic.

$(ii)$ Apply Proposition \ref{mub:lim}$(i)$ with
$\zeta$: 
$$
|\zeta_0[\sigma^k]+\dots+\zeta_d[\sigma^{k+d}]|=
|\zeta_0([\sigma^k]-\sigma^k)+\dots
+\zeta_d([\sigma^{k+d}]-\sigma^{k+d})|
\le\sum|\zeta_i|.\eqno{\eck}
$$






\subsection{Polynomial sequences}\index{polynomial sequences}





Let us first give some numerical evidence for the classical case of
sets of $d$th powers. The table below reads as 
follows: ``the set $E=\{k^d\}$ for $d$ the value in the first column
fails the property in the second column by the counterexample given in
the third column.'' Indeed, each such counterexample to $n$-independence
yields arbitrarily large counterexamples.\vskipb

\hfill\vbox{\halign{
\vrule\hbox to 2pt{}\lower4pt\vbox to 15pt{}#\hfil\hbox to 1pt{}\vrule&
\hfil#\hfil\vrule&\ #\hfil\hbox to 1pt{}\vrule\cr
\noalign{\hrule}
$\{k^d\}$&\ fails\ &by counterexample\cr
\noalign{\hrule}
%\noalign{\vskip1pt\hrule}
d=2\index{polynomial sequences!squares}&\I{2}& $7^2+1^2=2\cdot5^2$ 
(or $18^2+1^2=15^2+10^2$ \cite[book II, problem 9]{di59})\cr
d=3\index{polynomial sequences!cubes}&\I{2}& $12^3+1^3=10^3+9^3$ 
\cite[due to Fr\'enicle]{br96}\index{Fr\'enicle de Bessy, Bernard}\cr
d=4\index{polynomial sequences!biquadrates}&\I{2}& 
$158^4+59^4=134^4+133^4$ 
(or $12231^4+2903^4=10381^4+10203^4$ \cite{eu72})\cr
d=5&\I{3}& $67^5+28^5+24^5=62^5+54^5+3^5$ 
(another first in \cite{mo39})\cr
d=6&\I{3}& $23^6+15^6+10^6=22^6+19^6+3^6$ \cite{ra34}\cr
d=7&\I{4}& $149^7+123^7+14^7+10^7=146^7+129^7+90^7+15^7$ \cite{ek96}\cr
d=8&\I{5}& $43^8+20^8+11^8+10^8+1^8=41^8+35^8+32^8+28^8+5^8$ (see \cite{ek98})\cr
d=9&\I{6}& $23^9+18^9+14^9+2\cdot13^9+1^9=22^9+21^9+15^9+10^9+9^9+5^9$
\cite{lps67}
\cr
d=10&\I{7}& $38^{10}+33^{10}+2\cdot26^{10}+15^{10}+8^{10}+1^{10}=$\cr
&&\hfill$36^{10}+35^{10}+32^{10}+29^{10}+24^{10}+23^{10}+22^{10}$
(another first in \cite{mo39})\cr 
\noalign{\hrule}
}}\hfill\hbox{}\nopagebreak\smallskip\\\nopagebreak
\hfill\refstepcounter{thmsub}Table \thethmsub\hfill\hbox{}


\vskipa 
Note that a positive answer to Euler's\index{Euler's conjecture}
conjecture
---~for $k\ge5$ $a^k+b^k=c^k+d^k$ has only
trivial solutions
in integers~--- would imply that
the set of $k$th powers has \I{2}. This
conjecture has been neither proved nor disproved for any
value of $k\ge 5$ (see \cite{sw95} and \cite{ek98}).

Now let \DEE/ be a set of polynomial growth: $|n_k|\preccurlyeq k^d$ for
some $d\ge1$. Then $\mes{E\cap[-n,n]}\succcurlyeq n^{1/d}$ and by
\cite[Th.\ 3.6]{ru60}, $E$ fails the \EL{p} property for $p>2d$
and $E$ fails {\it a fortiori}\/ \I{d+1}. In the special case
$E=\{P(k)\}$ for a polynomial $P$ of degree $d$, we can exhibit a huge
explicit arithmetical relation. 
%
%
%
%Let $E=\{P(k)\}$ for a polynomial $P$ of degree $d$.
%
%As usual, the investigation becomes more number
%theoretical, but certain large arithmetical relations
%necessarily hold. 
Recall that
%
\begin{equation}\label{mub:poly}
%
\Delta^jP(k)=\suml_{i=0}^j\bi ji(-1)^iP(k-i)
\ ,\ \suml_{i=0}^j\bi ji(-1)^i=0
\ ,\ \suml_{i=0}^j\bi ji=2^j.
%
\end{equation}
%
As $\Delta^{d+1}P(k)=0$, this makes $E$ fail \I{2^d}, which is coarse. 
\vskipb

{\bf Conclusion }
By Theorem \ref{mub:thm},
property \I{n} yields directly \umbs/ in the spaces $\SL{2p}{}$, $p\le n$ 
integer.
But we do not know whether
\I{\infty} ensures \umbs/ in spaces
$\SLP{}$, $p$ not an even integer.









\section[Metric unconditional approximation property \umap/]
        {Metric unconditional approximation property}
\label{sect:block}









As we investigate 
simultaneously real and complex \umap/, 
it is convenient to introduce a 
subgroup $\U$ of $\T$ corresponding to each case. Thus, 
if $\U=\D=\{-1,1\}$, then 
the following applies to real \umap/. 
If $\U=\T=\{\epsilon\in\C:|\epsilon|=1\}$, it applies to complex \umap/.

He who is first and foremost interested in the 
application to harmonic analysis may concentrate on the equivalence 
$(ii)\Leftrightarrow(iv)$ in Theorem \ref{block:thm} and then 
pass on to Section \ref{sect:tis}.





\subsection{Definition}







We start with defining the metric unconditional 
approximation property (\umap/ for short). Recall that 
$\Delta T_k=T_k-T_{k-1}$ (where $T_0=0$).
%
\begin{dfnsub}\label{block:def}\index{approximation property}
%
  Let $X$ be a separable Banach space.
  
  \bm{i} A sequence $\{T_k\}$ of operators on $X$ is an approximating
  sequence\index{approximating sequence} \(\as/ for short\) if each
  $T_k$ has finite rank and $\|T_kx-x\|\to0$ for every $x\in X$.  If
  $X$ admits an \as/, it has the bounded approximation property.  An
  \as/ of commuting projections is a finite-dimensional
  decomposition\index{finite-dimensional decomposition} \(\fdd/ for
  short\).
  
  \bm{ii} \cite{fe80} $X$ has the unconditional approximation property
  \index{unconditional approximation property} \uap/ if there are an
  \as/ $\{T_k\}$ and a constant $C$ such that
%
\begin{equation}\label{block:uapdef}
%
\biggl\|\sum_{k=1}^n\epsilon_k\Delta T_k\biggr\|
\le C\quad\hbox{for all $n$ and $\epsilon_k\in\U$}.
%
\end{equation}
%
The \uap/ constant is the least such $C$.

\bm{iii} \cite[\S3]{ck91} $X$ has the metric unconditional
approximation 
property\index{metric unconditional approximation property} \umap/ if it has \uap/ with constant $1+\eps$ for any
$\eps>0$.
%
\end{dfnsub}
%
Property $(ii)$ is the approximation property which most 
appropriately generalizes the 
unconditional basis property. It has first been 
introduced by Pe\l czy\'nski \index{Pe\l czy\'nski, Aleksander} and 
Wojtaszczyk\index{Wojtaszczyk, Przemys\l aw} \cite{pw71}. 
They showed that it holds if and only if $X$ is a complemented 
subspace of a space with an unconditional 
\fdd/\index{unconditional fdd@unconditional \fdd/}. 
By 
\cite[Th.\ 1.g.5]{lt77}, 
this implies that $X$ is subspace of a space with an 
unconditional basis. Thus, neither ${\fam0 L}^1([0,1])$ nor 
$\script{C}([0,1])$ share \uap/.
 
Property $(iii)$ has been introduced by 
Casazza\index{Casazza, Peter G.} and 
Kalton\index{Kalton, Nigel J.} as an 
extreme form of metric approximation. It has been studied in 
\cite[\S3]{ck91}, 
\cite[\S8,9]{gks93}, \cite{gkl96} and \cite[\S IV]{gk97}.

There is a simple and very useful criterion for \umap/: 
%
\begin{prpsub}
  [\protect{\cite[Th.\ 3.8]{ck91} and \cite[Lemma 8.1]{gks93}}]
\label{block:umap:eq}
%
Let $X$ be a separable Banach space.  $X$ has \umap/ if and only if
there is an \as/ $\{T_k\}$ such that
%
\begin{equation}\label{block:umap:def}
%
\sup_{\epsilon\in\sU}\|(\Id-T_k)+\epsilon T_k\|\tol_{k\to\infty}1.
%
\end{equation}
%
\end{prpsub}
%
If \Ref{block:umap:def} holds, we say that $\{T_k\}$ realizes 
\umap/. A careful reading of the above mentioned proof 
also gives 
the following results for \as/ 
that satisfy $T_{n+1}T_n=T_n$. 
%
\begin{prpsub}
%
  Let $X$ be a separable Banach space.
  
  \bm{i} Let $\{T_k\}$ be an \as/ for $X$ such that $T_{n+1}T_n=T_n$.
  A subsequence $\{T'_k\}$ of $\{T_k\}$ realizes $1$-\uap/ in $X$ if
  and only if for all $k\ge1$ and $\epsilon\in\U$
  $$
    \|\Id-(1+\epsilon)T'_k\|=1.
  $$
  
  \bm{ii} $X$ has metric unconditional 
\fdd/\index{metric unconditional fdd@metric unconditional \fdd/} if and only if there
  is an \fdd/ $\{T_k\}$ such that $(\ref{block:umap:def})$ holds.
%
\end{prpsub}















\subsection{Characterization of \umap/. Block unconditionality}













We want to characterize \umap/ in an even simpler 
way than Proposition \ref{block:umap:eq}. 
Relation $(\ref{block:umap:def})$ and 
the method of \cite[Th.\ 4.2]{kw95}, suggest 
considering some unconditionality condition between 
a certain ``break'' and a certain ``tail'' of $X$. We 
propose two such notions. 
%
\begin{dfnsub}\label{block:def:u}
%
  Let $X$ be a separable Banach space.
  
  \bm{i} Let $\tau$ be a vector space topology on $X$.  Then $X$ has
  the property $(u(\tau))$ of
  $\tau$-unconditionality\index{tau-unconditionality@$\tau$-unconditionality}
  if for all $u\in X$ and norm bounded sequences $\{v_j\}\se X$ such
  that $v_j\hoch{\to}^{\tau}0$
\begin{equation}\label{block:def:u:1}
\osc_{\epsilon\in\sU}\|\epsilon u+v_j\|\to0.
\end{equation}

\bm{ii} Let $\{T_k\}$ be a commuting a.s.  $X$ has the property
$(u(T_k))$ of commuting block 
unconditionality\index{commuting block unconditionality} if for all $\eps>0$ and $n\ge1$ we may choose
$m\ge n$ such that for all $x\in T_nB_X$ and $y\in (\Id-T_m)B_X$
%
\begin{equation}\label{block:def:u:2}
%
\osc_{\epsilon\in\sU}\|\epsilon x+y\|\le\eps.
%
\end{equation}
%
\end{dfnsub}
%
Thus, given a commuting \as/ 
$\{T_k\}$, $T_nX$ 
is the ``break''\index{break} and $(\Id-T_m)X$ the
``tail''\index{tail} of $X$. 
%If the sequence is not commuting, there are more
%complex notions that accurately describe break and tail: see the
%definition of $A_n$ in the proof of \cite[Th.\ 4.2]{kw95}.
We have 
%
\begin{lemsub}\label{block:lm}
%
  Let $X$ be a separable Banach space and $\{T_k\}$ a commuting \as/
  for $X$. The following are equivalent.
  
  \bm{i} $X$ enjoys $(u(\tau))$ for some vector space topology $\tau$
  such that $T_nx\hoch{\to}^{\tau}x$ uniformly for $x\in B_X$;
  
  \bm{ii} $X$ enjoys $(u(T_k))$.
%
\end{lemsub}
%
\dem
%
$(i)\imp(ii)$. 
Suppose that $(ii)$ fails: there are 
$n\ge1$ and $\eps>0$ such that for each $m\ge n$, there 
are $x_m\in T_nB_X$
and $y_m\in(\Id-T_m)B_X$ 
such that 
$$
\osc_{\epsilon\in\sU}\|\epsilon x_m+y_m\|>\eps.
$$
As $T_nB_X$ is compact, we may suppose by extracting 
a convergent subsequence that $x_m=x$. 
Let $\tau$ be as in $(i)$: then $y_m\hoch{\to}^{\tau}0$ 
and $(u(\tau))$ must fail.

$(ii)\imp(i)$. 
Let us define a vector space topology $\tau$ 
by 
$$
x_n\hoch{\to}^\tau0\quad\Longleftrightarrow\quad
\forall k\ \|T_kx_n\|\to0.
$$
Then $T_nx\hoch{\to}^\tau x$ uniformly on $B_X$. 
Indeed, $T_k(T_nx-x)=(T_n-\Id)T_kx$ and $T_n-\Id$ converges 
uniformly to 0 on $T_kB_X$ which is norm compact.

Let us 
check $(u(\tau))$. Let $u\in B_X$ and 
$\{v_j\}\se B_X$ be such 
that $v_j\hoch{\to}^{\tau}0$. 
Let $\eps>0$. There is $n\ge1$ such that 
$\|T_nu-u\|\le\eps$. 
Choose $m$ such that $(\ref{block:def:u:2})$ holds 
for $x\in T_nB_X$ and $y\in (\Id-T_m)B_X$.
Then choose $k\ge1$ such that 
$\|T_mv_j\|\le\eps$ for 
$j\ge k$. We have, for any $\epsilon\in\U$,
%
\begin{eqnarray*}
%
\|\epsilon u+v_j\|&\le&
\|\epsilon T_nu+(\Id-T_m)v_j\|+\|T_nu-u\|+\|T_mv_j\|\\
&\le&\|T_nu+(\Id-T_m)v_j\|+3\eps\le\|u+v_j\|+5\eps.
%
\end{eqnarray*}
%
Thus we have $(\ref{block:def:u:1})$.
%
\eck 


In order to obtain \umap/ from block independence, we shall
have to construct unconditional 
skipped blocking decompositions. 
%
\begin{dfnsub}\label{block:sbddef}
%
  Let $X$ be a separable Banach space. $X$ admits unconditional
  skipped blocking 
decompositions\index{unconditional skipped blocking decompositions} if for each $\eps>0$, there is an unconditional
  \as/ $\{S_k\}$ such that for all $0\le a_1<b_1<a_2<b_2<\dots$ and
  $x_k\in(S_{b_k}-S_{a_k})X$
  $$
  \sup_{\epsilon_k\in\sU}\Bigl\|\sum
  \epsilon_kx_k\Bigr\|\le(1+\eps)\Bigl\|\sum x_k\Bigr\|.
  $$
%
\end{dfnsub}






\subsection{Main theorem: convex combinations of multipliers}








We have 
%
\begin{thmsub}\label{block:thm}
  \index{metric unconditional approximation property} Consider the
  following properties for a separable Banach space $X$.
  
  \bm{i} There are an unconditional commuting \as/ $\{T_k\}$ and a
  vector space topology $\tau$ such that $X$ enjoys $(u(\tau))$ and
  $T_kx\hoch{\to}^{\tau}x$ uniformly for $x\in B_X$;
  
  \bm{ii} $X$ enjoys $(u(T_k))$ for an unconditional commuting \as/
  $\{T_k\}$;
  
  \bm{iii} $X$ admits unconditional skipped blocking decompositions;
  
  \bm{iv} $X$ has \umap/.
  
  Then $(iv)\imp(i)\Leftrightarrow(ii)\Rightarrow(iii)$.  If $X$ has
  finite cotype, then $(iii)\imp(iv)$.
%
\end{thmsub}
%
\dem
%
$(i)\Leftrightarrow(ii)$ holds by Lemma \ref{block:lm}.

$(iv)\imp(ii)$. By 
Godefroy\index{Godefroy, Gilles}--Kalton's\index{Kalton, Nigel J.} 
\cite[Th.\ IV.1]{gk97}, there 
is in fact an \as/ $\{T_k\}$ that satisfies 
$(\ref{block:umap:def})$ such that 
$T_kT_l=T_{\min(k,l)}$ if $k\ne l$. 

Let $C$ be a uniform bound for $\|T_k\|$. 
Let $\eps>0$ and $n\ge1$. There is $m\ge n+2$ such that 
$$
\sup_{\epsilon\in\sU}
\|\epsilon T_{m-1}+(\Id-T_{m-1})\|\le1+\eps/2C.
$$ 
Let $x\in T_nB_X$ and $y\in(\Id-T_m)B_X$. As $x-T_{m-1}x=0$ and 
$T_{m-1}y=0$,
$$
\epsilon x+y=\epsilon T_{m-1}(x+y)+
(\Id-T_{m-1})(x+y),
$$
and, for all $\epsilon\in\U$, 
$$
\|\epsilon x+y\|\le(1+\eps/2C)\|x+y\|\le\|x+y\|+\eps.
$$

$(ii)\imp(iii)$. 
By a perturbation \cite[proof of Lemma III.9.2]{si81}, we may
suppose that $T_kT_l=T_{\min(k,l)}$ if $k\ne l$. 
Let $\eps>0$ and choose a sequence of $\eta_j>0$ 
such that $1+\eps_j=\prod_{i\le j}(1+\eta_i)<1+\eps$ 
for all $j$. 
By $(ii)$, there is a subsequence $\{S_j=T_{k_j}\}$
such that $k_0=0$ and thus $S_0=0$, and 
%
\begin{equation}\label{block:h}
%
\sup_{\epsilon\in\sU}\|x+\epsilon y\|\le(1+\eta_j)\|x+y\|
\end{equation}
for $x\in(\Id -S_j)X$ and $y\in S_{j-1}X$. Let us show that 
it is an unconditional skipped blocking decomposition: we 
shall prove by induction that
%
$$
\left\{\!\!\begin{array}{l}
\ds\sup_{\epsilon_i\in\sU}
\biggl\|x+\sum_{i=1}^n
\epsilon_ix_i\biggr\|\le(1+\eps_j)
\biggl\|x+\sum_{i=1}^nx_i\biggr\|\mbox{ for $x\in(\Id -S_j)X$}\\
\mbox{and $x_i\in(S_{b_i}-S_{a_i})X$ 
$(0\le a_1<b_1<\dots<a_n<b_n\le j-1)$.}
\end{array}\right.\leqno{(H_j)}
$$

\bloc $(H_1)$ trivially holds.

\bloc 
Assume $(H_i)$ holds for $i<j$. Let $x$ and $x_i$ as in $(H_j)$. 
Let $\epsilon_i\in\U$. Then
$$
\Bigl\|x+\sum_{i=1}^n\epsilon_ix_i\Bigr\|\le
(1+\eta_j)\Bigl\|x+\overline{\epsilon_n}
\sum_{i=1}^n\epsilon_ix_i\Bigr\|=
(1+\eta_j)\Bigl\|x+x_n+
\sum_{i=1}^{n-1}\overline{\epsilon_n}\epsilon_ix_i\Bigr\|
$$
by $(\ref{block:h})$. Note that $x+x_n\in(\Id -S_{a_n})X$: 
an application of $(H_{a_n})$ yields $(H_j)$.

$(iii)\imp(iv)$. 
Let $\eps>0$, $n>1$. 
There is an unconditional skipped blocking 
decomposition $\{S_k\}$. Let $C_u$ be the \uap/ constant 
of $\{S_k\}$. Let 
$$
V_{i,j}=S_{in+j-1}-S_{(i-1)n+j}\qquad
\mbox{for $1\le j\le n$ and $i\ge 0$.}
$$
The $j$th skipped blocks are 
$$
U_j=\Id -\sum_i V_{i,j}=\sum_i\Delta S_{in+j};
$$
then 
$\sum_{j=1}^n U_j=\Id $. Let 
$$
R_i=\frac1{n-1}\sum_{j=1}^n V_{i,j};
$$
then $R_i$ has finite rank and 
$$
R_0+R_1+\dots=(n\Id -\Id )/(n-1)=\Id .
$$
Thus $W_j=\sum_{i\le j}R_i$ defines an a.s.
We may bound its \uap/ constant. First, since 
$\{S_k\}$ is a skipped blocking decomposition,
%
\begin{eqnarray*}
%
\forall x\in B_X\quad\sup_{\epsilon_i\in\sU}
\Bigl\|\sum\epsilon_iR_ix\Bigr\|
&\le&\frac1{n-1}\sum_{j=1}^n\sup_{\epsilon_i\in\sU}
\Bigl\|\sum_i\epsilon_iV_{i,j}x\Bigr\|\\
&\le&\frac{1+\eps}{n-1}\sum_{j=1}^n\|x-U_jx\|\\
&\le&\frac{1+\eps}{n-1}\Bigl(n+\sum_{j=1}^n\|U_jx\|\Bigr).
%
\end{eqnarray*}
%
Let us bound $\sum_1^n\|U_jx\|$. 
Let $q<\infty$ be the cotype of $X$ and 
$C_c$ its cotype constant. 
Then by H\"older's inequality we have for all $x\in B_X$
\begin{eqnarray}\label{block:cotype}
%\forall x\in B_X\quad
\nonumber\sum\|U_jx\|
&\le&n^{1-1/q}
\Bigl(\sum
\|U_jx\|^q\Bigr)^{1/q}\\
&\le&n^{1-1/q}C_c\cdot\mathop{\hbox{average}}\limits_{\pm}
\Bigl\|\sum\pm U_jx\Bigr\|
\le n^{1-1/q}C_cC_u.
\end{eqnarray}
Thus the \uap/ constant of 
$\{W_j\}$ is at most
$(1+\eps)(n+C_cC_un^{1-1/q})/(n-1)$.
As $\eps$ is arbitrarily little and $n$ arbitrarily 
large, $X$ has \umap/.\eck\vskipb

\remsub
How does Theorem \ref{block:thm} look in the special cases where $\tau$ is
the weak or the weak$^*$ topology ? They correspond to the classical
cases where the \as/ is 
shrinking\index{shrinking approximating sequence} \vs boundedly 
complete\index{boundedly complete approximating sequence}.\vskipb
 
We may remove the cotype\index{cotype} assumption in 
Theorem \ref{block:thm} 
$(iii)\imp(iv)$ if the space has the properties of 
commuting \lap{1} or $\ell_q$-\fdd/ for $q<\infty$, 
which will be introduced in Section \ref{sect:lpmap}:
%
\begin{thmsub}\label{block:thm'}
%
  Consider the following properties for a separable Banach space $X$.
  
  \bm{i} There are a commuting $\ell_1$-\as/ or an $\ell_q$-\fdd/
  $\{T_k\}$, $q<\infty$, and a vector space topology $\tau$ such that
  $X$ enjoys $(u(\tau))$ and $T_kx\hoch{\to}^{\tau}x$ uniformly for
  $x\in B_X$;
  
  \bm{ii} $X$ enjoys $(u(T_k))$ for a commmuting $\ell_1$-\as/ or an
  $\ell_q$-\fdd/ $\{T_k\}$, $q<\infty$;
  
  \bm{iii} $X$ admits unconditional skipped blocking decompositions
  and one may in fact take an $\ell_1$-\as/ or an $\ell_q$-\fdd/
  $\{T_k\}$, $q<\infty$, in its definition \ref{block:sbddef};
  
  \bm{iv} $X$ has \umap/.
  
  Then $(i)\Leftrightarrow(ii)\imp(iii)\imp(iv)$.
%
\end{thmsub}
%
\dem
%
Part $(i)\Leftrightarrow(ii)\imp(iii)$ goes as before. 
To prove $(iii)\imp(iv)$, note that in the proof of 
Theorem \ref{block:thm} 
$(iii)\imp(iv)$, one may replace 
the estimate in $(\ref{block:cotype})$ by 
$$
\forall x\in B_X\quad\sum\|U_jx\|\le 
n^{1-1/q}\Bigl(\sum
\|U_jx\|^q\Bigr)^{1/q}\le n^{1-1/q}C_{\ell},
$$
where $C_{\ell}$ is the $\ell_1$-\ap/ or
the $\ell_q$-\fdd/ constant.
%
\eck












\section{The $p$-additive approximation property \lpap/}
\label{sect:lpmap}






\subsection{Definition}







\begin{dfnsub}\label{str:def}
%
  Let $X$ be a separable Banach space.
  
  \bm{i} $X$ has the $p$-additive approximation property
  \lpap/\index{p-additive approximation property@$p$-additive approximation property} if there are an \as/ $\{T_k\}$ and a
  constant $C$ such that
%
\begin{equation}\label{block:def:str}
%
C^{-1}\|x\|\le
\Bigl(\sum\|\Delta T_kx\|^p\Bigr)^{1/p}\le C\|x\|
%
\end{equation}
%
for all $x\in X$. The \lpap/ constant is the least such $C$.

\bm{ii} $X$ has the metric $p$-additive
approximation 
property\index{metric $p$-additive approximation property} 
\lpmap/ if it has \lpap/ with constant $1+\eps$ for any
$\eps>0$.
%
\end{dfnsub}
%
Note that \lpap/ implies \uap/ and
\lpmap/ implies \umap/. 
Note also that in $(\ref{block:def:str})$,
the left inequality is trivial with $C=1$ if $p=1$; 
the right inequality is always achieved for some $C$ if $p=\infty$. 

Property $(ii)$ is implicit in 
Kalton\index{Kalton, Nigel J.}--Werner's\index{Werner, Dirk}
\cite{kw95} investigation of subspaces of ${\fam0 L}^p$ that
are almost isometric to subspaces of $\ell_p$: 
see Section \ref{sect:appen}.

The proof of Proposition \ref{block:umap:eq} 
can be adapted to yield
%
\begin{prpsub}\label{block:str:eq}
%
  Let $X$ be a separable Banach space.
  
  \bm{i} If there is an \as/ $\{T_k\}$ such that
\begin{equation}\label{block:umapf:def}
\Bigl(\|x-T_kx\|^p+\|T_kx\|^p\Bigr)^{1/p}
\tol_{k\to\infty}1
\end{equation}
uniformly on the unit sphere, then $X$ has \lpmap/.  The converse
holds if $p=1$.

\bm{ii} $X$ has a metric $\ell_p$-\fdd/ if and only if there is an
\fdd/ $\{T_k\}$ such that $(\ref{block:umapf:def})$ holds.
%
\end{prpsub}
%
We shall say that
$\{T_k\}$ realizes \lpmap/
if it satisfies
$(\ref{block:umapf:def})$.\vskipb

\dem
Let $\{T_k\}$ be an \as/ that
satisfies $(\ref{block:umapf:def})$ and $\eps>0$.
By a perturbation \cite[Lemma 2.4]{jrz71},
we may suppose that $T_{k+1}T_k=T_k$.
Choose a sequence of $\eta_j>0$ such that
$1+\eps_k=\prod_{j\le k}(1+\eta_j)\le1+\eps$
for each $k$.
We may assume by taking a subsequence of the $T_k$'s that
for all $k$ and $x\in X$,
\begin{equation}\label{block:equ:dem}
(1+\eta_k)^{-1}\|x\|\le
\Bigl(\|x-T_kx\|^p+\|T_kx\|^p\Bigr)^{1/p}
\le(1+\eta_k)\|x\|.
\end{equation}
We then prove by induction the hypothesis $(H_k)$
$$
\forall x\in X\quad
(1+\eps_k)^{-1}\|x\|\le
\Bigl(\|x-T_kx\|^p+
\sum_{j=1}^k\|\Delta T_jx\|^p\Bigr)^{1/p}
\le(1+\eps_k)\|x\|.
$$

\bloc
$(H_1)$ is true.

\bloc
Suppose $(H_{k-1})$ is true. Let $x\in X$. Note
that
$$
x-T_kx=(\Id -T_k)(x-T_{k-1}x)\quad,\quad
\Delta T_kx=T_k(x-T_{k-1}x).
$$
By $(\ref{block:equ:dem})$, we get
\begin{eqnarray*}
\bigl(\|x-T_kx\|^p+\|\Delta T_kx\|^p\bigr)^{1/p}
\le(1+\eta_k)\|x-T_{k-1}x\|.
\end{eqnarray*}
Hence, by $(H_{k-1})$,
\begin{eqnarray*}
\lefteqn{\Bigl(
\|x-T_kx\|^p+\sum_{j=1}^k\|\Delta T_jx\|^p
\Bigr)^{1/p}\le}\quad\quad\quad&&\\
\quad\quad&\le&(1+\eta_k)\Bigl(\|x-T_{k-1}x\|^p+
\sum_{j=1}^{k-1}\|\Delta T_jx\|^p\Bigr)^{1/p}
\le(1+\eps_k)\|x\|.
\end{eqnarray*}


\bloc
We obtain the lower bound in the same way.
Thus the induction is complete.

Hence $\{T_k\}$ realizes
\lpap/ with constant $1+\eps$. As $\eps$ is arbitrary, 
$X$ has \lpmap/.

If $X$ has \lmap{1}, then for each $\eps>0$,
there is a sequence $\{S_k\}$ such that
$$
\|x\|\le\|x-S_kx\|+\|S_kx\|\le
\sum\|\Delta S_kx\|\le(1+\eps)\|x\|
$$
for all $x\in X$. By a diagonal
argument, this gives an \as/
$\{T_k\}$ satisfying $(\ref{block:umapf:def})$.

$(iii)$
If $X$ has a metric $\ell_p$-\fdd/,
then for each $\eps>0$ there is a 
\fdd/ $\{T_k\}$ such that
$(\ref{block:def:str})$ holds with $C=1+\eps$.
Then, for all $k\ge1$,
$$
(1-\eps)\|T_kx\|\le\biggl(\sum_{j=1}^k\|\Delta T_jx\|^p
\biggr)^{1/p}\le(1+\eps)\|T_kx\|
$$
$$
(1-\eps)\|x-T_kx\|\le
\biggl(\sum_{j=k+1}^\infty\|\Delta T_jx\|^p\biggr)^{1/p}
\le(1+\eps)\|x-T_kx\|.
$$
Thus
$$
(1-\eps)/(1+\eps)\|x\|\le
\Bigl(\|x-T_kx\|^p+\|T_kx\|^p\Bigr)^{1/p}
\le(1+\eps)/(1-\eps)\|x\|.
$$
By a diagonal
argument, this gives an \fdd/
$\{T_k\}$ satisfying $(\ref{block:umapf:def})$.
\eck\vskipb

\questsub
What about the converse in Proposition \ref{block:str:eq}$(i)$ for
$p>1$~? 









\subsection{Some consequences of \lpap/}









We start with the simple
%
\begin{prpsub}\label{appen:l1}
%
  Let $X$ be a separable Banach space.
  
  \bm{i} If $X$ has \lpap/ with constant $C$, then $X$ is
  $C$-isomorphic to a subspace of an $\ell_p$-sum of finite
  dimensional subspaces of $X$.
  
  \bm{ii} If furthermore $X$ is a subspace of ${\fam0 L}^q$, then $X$ is
  $(C+\eps)$-isomorphic to a subspace of $(\bigoplus \ell_q^n)_p$ for
  any given $\eps>0$.
  
  \bm{iii} In particular, if a subspace of ${\fam0 L}^p$ has \lpap/ with
  constant $C$, then it is $(C+\eps)$-isomorphic to a subspace of
  $\ell_p$ for any given $\eps>0$. If a subspace of ${\fam0 L}^p$ has \lpmap/,
  then it is almost isometric to subspaces of $\ell_p$.
%
\end{prpsub}
%
\dem
%
$(i)$
Indeed, $\Phi\colon X\hookrightarrow(\bigoplus\im\Delta T_i)_{p}$, 
$x\mapsto\{\Delta T_ix\}_{i\ge1}$
is an embedding: for all $x\in X$
$$
C^{-1}\|x\|_X\le\|\Phi x\|=
\Bigl(\sum\|\Delta T_ix\|^p_X\Bigr)^{1/p}\le C\|x\|_X.
$$

$(ii\,\&\,iii)$
Recall that, given $\eps>0$, a
finite dimensional subspace of ${\fam0 L}^q$
is $(1+\eps)$-isomorphic to a subspace
of $\ell_q^n$ for some $n\ge1$.
%
\eck\vskipb

We have in particular
(see \cite[\S VIII, Def.\ 7]{hmp86}
for the definition of Hilbert sets)
%
\begin{corsub}\label{appen:uap}
  
  Let \DE/ be infinite.
  
  \bm{i} No $\SLE{q}$ 
\index{p-additive approximation property@$p$-additive approximation property!for spaces $\SLPE$}
  $(1\le q<\infty)$ has \lpap/ for $p\ne2$.
  
  \bm{ii} No $\SCE$ 
\index{p-additive approximation property@$p$-additive approximation property!for spaces $\SCE$}
  has \lap{q} for $q\ne1$.  If $E$ is a Hilbert\index{Hilbert set}
  set, then $\SCE$ fails \lap{1}.
%
\end{corsub}
%
\dem
%
This is a consequence of Proposition \ref{appen:l1}$(i)$: every
infinite $E$ contains a Sidon set and
thus a \EL{2\vee p} set. So $\SLPE$ contains $\ell_2$.
Also, if $E$ is a Hilbert set, then $\SCE$ contains $c_0$
by \cite[Th.\ 2]{li95}.
%
\eck\vskipb

However, there is a Hilbert set $E$ such that
$\SCE$ has complex \umap/: see \cite[Th.\ 10]{li96}.
The class of sets $E$ such that $\SCE$ has \lap{1}
contains the Sidon\index{Sidon set} sets and 
Blei's\index{Blei, Ron C.} 
sup-norm-partitioned sets\index{sup-norm-partitioned sets} 
\cite{bl74}. 

















\subsection{Characterization of \lpmap/}









Recall \cite[Def.\ 4.1]{kw95}: 
%
\begin{dfnsub}\label{block:strong:dfn}
%
  Let $X$ be a separable Banach space.
  
  \bm{i} Let $\tau$ be a vector space topology on $X$. $X$ enjoys
  property $(m_p(\tau))$ if for all $x\in X$ and norm bounded
  sequences $\{y_j\}$ such that $y_j\hoch{\to}^{\tau}0$
  $$
  \bigl| \|x+y_j\|-\bigl(\|x\|^p+\|y_j\|^p\bigr)^{1/p} \bigr|\to0.
  $$
  
  \bm{ii} $X$ enjoys the property $(m_p(T_k))$ for a commuting \as/
  $\{T_k\}$ if for all $\eps>0$ and $n\ge1$ we may choose $m\ge n$
  such that for all $x\in B_X$
  $$
  \bigl|\|T_nx+(\Id -T_m)x\|- \bigl(\|T_nx\|^p+\|(\Id
  -T_m)x\|^p\bigr)^{1/p} \bigr|\le\eps.
  $$
%
\end{dfnsub}
%
Then \cite[Th.\ 4.2]{kw95} may be read as follows 
%
\begin{thmsub}\label{block:strong:thm}
  \index{metric $p$-additive approximation property}
%
  Let $1\le p<\infty$ and consider the following properties for a
  separable Banach space $X$.
  
  \bm{i} There are an unconditional commuting \as/ $\{T_k\}$ and a
  vector space topology $\tau$ such that $X$ enjoys $(m_p(\tau))$ and
  $T_kx\hoch{\to}^{\tau}x$ uniformly for $x\in B_X$;
  
  \bm{ii} $X$ enjoys the property $(m_p(T_k))$ for an unconditional
  commuting \as/ $\{T_k\}$.
  
  \bm{iii} $X$ has \lpmap/.
  
  Then $(i)\Leftrightarrow(ii)$.  If $X$ has finite cotype, then
  $(ii)\imp(iii)$.
%
\end{thmsub}
%
As for Theorem \ref{block:thm}, we may remove
the cotype assumption if $X$ has commuting $\ell_1$-\ap/ or
$\ell_p$-\fdd/, $p<\infty$:
%
\begin{thmsub}\label{block:strong:thm'}
%
  Let $1\le p<\infty$.  Consider the following properties for a
  separable Banach space $X$.
  
  \bm{i} There are an $\ell_p$-\fdd/ \(or just a commuting
  $\ell_1$-\as/ in the case $p=1$\) $\{T_k\}$ and a vector space
  topology $\tau$ such that $X$ enjoys $(m_p(\tau))$ and
  $T_kx\hoch{\to}^{\tau}x$ uniformly for $x\in B_X$;
  
  \bm{ii} $X$ enjoys $(m_p(T_k))$ for an $\ell_p$-\fdd/ \(or just a
  commuting $\ell_1$-\as/ in the case $p=1$\) $\{T_k\}$.
  
  \bm{iii} $X$ has \lpmap/.
  
  Then $(i)\Leftrightarrow(ii)\imp(iii)$.
%
\end{thmsub}








\subsection{Subspaces of ${\fam0 L}^p$ with \lpmap/}
\label{sect:appen}








Although no translation invariant subspace of
$\SL{p}{}$ has \lap{p} for $p\ne2$, Proposition \ref{appen:l1}
$(iii)$ is not void. By the work of 
Godefroy\index{Godefroy, Gilles},
Kalton\index{Kalton, Nigel J.}, Li\index{Li, Daniel} and
Werner\index{Werner, Dirk}
\cite{kw95,gkl96}, we get examples of subspaces of
${\fam0 L}^p$ with \lmap{p} and even a
characterization of such spaces.

Let us treat the case $p=1$.
Recall first that a space $X$ has the
$1$-strong Schur\index{Schur property, $1$-strong} property
when, given $\delta\in{\mathopen]}0,2]$
and $\eps>0$, any
normalized $\delta$-separated sequence in $X$ contains
a subsequence that is $(2/\delta+\eps)$-equivalent to the
unit vector basis of $\ell_1$ (see \cite{ro79}).
In particular, a gliding hump argument
shows that any subspace of $\ell_1$
shares this property. By
Proposition \ref{appen:l1}$(iii)$, a space $X$
with \lmap{1} also does. Now recall the main theorem
of \cite{gkl96}:\vskipa

{\bf Theorem }
{\it Let $X$ be a subspace of ${\fam0 L}^1$ with the approximation
property. Then the following properties are equivalent:

$(i)$
The unit ball of $X$ is compact and
locally convex in measure;

$(ii)$
$X$ has \umap/ and the $1$-strong Schur property;

$(iii)$
$X$ is $(1+\eps)$-isomorphic to a $w^*$-closed 
subspace $X_\eps$ of $\ell_1$ for any $\eps>0$.}\vskipb

We may then add to these three the fourth equivalent property\vskipb

\bm{iv}
\index{metric $1$-additive approximation property!for subspaces of ${\fam0 L}^1$} 
$X$ has \lmap{1}.\vskipb

\dem
%
We just showed that $(ii)$ holds
when $X$ has \lmap{1}. Now suppose we have $(iii)$
and let $\eps>0$.
Thus there is a quotient $Z$ of $c_0$ such that $Z^*$ has
the approximation property and $Z^*$ is
$(1+\eps)$-isomorphic
to $X$.

Let us show that any such $Z^*$ has \lmap{1}.
$Z$ has beforehand the metric approximation
property, with say
$\{R_n\}$, because $Z^*$ has it as a dual separable space. 
By \cite[\protect{Th.\ 2.2}]{gs88}, 
$\{R_n^*\}$ is a metric \as/ in $Z^*$. 
Let $Q$ be the canonical quotient map from $c_0$ onto $Z$.
Let $\{P_n\}$ be the sequence of projections associated
to the natural basis of $c_0$. Then
$\{P_n^*\}$ is also an \as/ in $\ell_1$. Thus
$$
\|P_n^*Q^*x^*-Q^*R_n^*x^*\|_{\ell_1}\to0\qquad\mbox{for any }
x^*\in Z^*.
$$
By Lebesgue's dominated convergence theorem
(see \cite[Th.\ 1]{ka74}), $QP_n-R_nQ\to0$ weakly in the space
$\script{K}(c_0,Z)$ of
compact operators from $c_0$ to $Z$. By Mazur's theorem,
there are 
convex combinations $\{C_n\}$ of
$\{P_n\}$ and $\{D_n\}$ of $\{R_n\}$ such that
$\|QC_n-D_nQ\|_{\script{L}(c_0,Z)}\to0$. Thus
\begin{equation}\label{append:prox}
\|C_n^*Q^*-Q^*D_n^*\|_{\script{L}(Z^*,\ell_1)}\to0.
\end{equation}
Furthermore $C_n^*:\ell_1\to\ell_1$ has the form
$C_n^*(x_1,x_2,\dots)=(t_1x_1,t_2x_2,\dots)$
with $0\le t_i\le1$. Therefore, defining $Q^*a=(a_1,a_2,\dots)$,
%
\begin{eqnarray}
%
\nonumber\lefteqn{\|C_n^*Q^*a\|_1+\|Q^*a-C_n^*Q^*a\|_1=}
\qquad&&\\
\nonumber&=&\|(t_1a_1,t_2a_2,\dots)\|_1+
\|((1-t_1)a_1,(1-t_2)a_2,\dots)\|_1\\
\label{appen:fin}
&=&\sum(|t_i|+|1-t_i|)|a_i|=\sum|a_i|=\|Q^*a\|_1.
%
\end{eqnarray}
%
As $\{D_n^*\}$ is
still an \as/ for $Z^*$,
$\{D_n^*\}$ realizes \lmap{1} in $Z^*$ by
$(\ref{appen:fin})$,
$(\ref{append:prox})$ and Proposition \ref{block:str:eq}$(i)$.

Thus $X$ has \lap{1} with constant $1+2\eps$.
As $\eps$ is arbitrary, $X$ has \lmap{1}.
%
\eck\vskipb

For $1<p<\infty$, we have similarly by \cite[Th.\ 4.2]{kw95}
%
\begin{prpsub}
\index{metric $p$-additive approximation property!for subspaces of ${\fam0 L}^p$}
%
  Let $1<p<\infty$ and $X$ be a subspace of ${\fam0 L}^p$ with the
  approximation property. The following are equivalent:
  
  \bm{i} $X$ is $(1+\eps)$-isomorphic to a subspace $X_\eps$ of
  $\ell_p$ for any $\eps>0$.
  
  \bm{ii} $X$ has \lpmap/.
%
\end{prpsub}
%
\dem
$(ii)\imp(i)$ is in Proposition \ref{appen:l1}. 
For $(i)\imp(ii)$, it suffices 
to prove that any subspace $Z$ of $\ell_p$ 
with the approximation property has \lpmap/.

As $Z$ is reflexive, $Z$ admits a commuting shrinking \as/
$\{R_n\}$. Let $i$ be the injection of $Z$ into $\ell_p$. Let
$\{P_n\}$ be the sequence of projections associated to the
natural basis of $\ell_p$. It is also an \as/ for $\ell_{p'}$. Thus
%
$$
\|i^*P_n^*x^*-R_n^*i^*x^*\|_{Z^*}\to0\qquad
\mbox{for any }x^*\in\ell_{p'}.
$$
As before,
there are convex combinations $\{C_n\}$ of
$\{P_n\}$ and $\{D_n\}$ of $\{R_n\}$ such that
$\|C_ni-iD_n\|\to0$. 
The convex combinations are finite and may be chosen not to
overlap, so that for each $n\ge1$ there is
$m>n$ such that 
$$
\|C_nx+(\Id -C_m)x\|=\bigl(\|C_nx\|^p+\|(\Id -C_m)x\|^p\bigr)^{1/p}
$$
for $x\in\ell_p$. 
Thus $Z$ satisfies the property $(m_p(D_n))$. 
Following the lines of \cite[Lemma 1]{fe80}, we observe that
$\{D_n\}$ is a commuting unconditional \as/ since $\{P_n\}$ is.
By Theorem \ref{block:strong:thm}, $Z$
has \lpmap/.\eck










\section{\uap/ and \umap/ 
in translation invariant spaces}\label{sect:tis}














Recall that $\U$ is a subgroup of $\T$. 
If $\U=\D=\{-1,1\}$, the following applies to real \umap/. 
If $\U=\T=\{\epsilon\in\C:|\epsilon|=1\}$, it applies to complex
\umap/. 







\subsection{General properties. Isomorphic case}







$\SLP{}$ spaces $(1<p<\infty)$ are known to have an 
unconditional basis; furthermore, they have an 
unconditional \fdd/ in translation invariant subspaces 
$\SLP{I_k}$: this 
is a corollary of Littlewood--Paley
%\index{Littlewood, John Edensor}
%\index{Paley, Raymond Edward Alan Christopher}
\index{Littlewood--Paley partition} 
theory \cite{lp31}. 
One may choose $I_0=\{0\}$ and 
$I_k=\mathopen]-2^k,-2^{k-1}]\cup
[2^{k-1},2^k\mathclose[$. Thus any 
$\SLPE$ $(1<p<\infty)$ has an unconditional \fdd/ 
in translation invariant subspaces 
$\SLP{E\cap I_k}$. The
spaces $\SL{1}{}$ and $\SC{}$, however, do not even have 
\uap/.
 
%Li \cite[Lemma 5]{li96} proves that in translation invariant 
%subspaces, \umap/ may as well be achieved 
%with multipliers of finite rank. His proof 
%applies also to \uap/, \lap{1} and \lmap{1}. Thus he shows
%
\begin{prpsub}[see \protect{\cite[Lemma 5, Cor.\ 6, Th.\ 7]
    {li96}}]\label{sbd:ssesp}
%
  Let $E\se\Z$ and $X$ be a homogeneous Banach space on \T.
  
  \bm{i} If $X_E$ has \umap/ \(\vs \uap/, \lap{1} or \lmap{1}\), then
  some \as/ of multipliers realizes it.
  
  \bm{ii} Let $F\se E$. If $X_E$ has \umap/ \(\vs \uap/, \lap{1} or
  \lmap{1}\), then so does $X_F$.
% also has \umap/ \(\vs \uap/, \lap{1} or \lmap{1}\).
  
  \bm{iii} If $\SCE$ has \umap/ \(\vs \uap/\),
then so does $X_E$.% all $\SLPE$ $(1\le p<\infty)$. If $\SCE$ has \uap/, 
%then so does $\SL{1}{E}$.
\end{prpsub}
%
Note the important property that \as/ of multipliers commute and 
commute with one another.

Whereas \uap/ is always satisfied for 
$\SLPE$ 
$(1<p<\infty)$, we have the following generalization of 
\cite[remark after Th.\ 7, Prop.\ 9]{li96} for the spaces 
$\SL{1}{E}$ and $\SCE$. By the method of \cite{gk97},
%
\begin{lemsub}\label{uap:sep}
%
  If $X$ has \uap/ with a commuting \as/ and $X\not\supseteq c_0$,
  then $X$ is a dual space.
%
\end{lemsub}
%
\dem
Suppose $\{T_n\}$ is a commuting \as/ such that 
$(\ref{block:uapdef})$ holds. As $X\not\supseteq c_0$, 
$Px^{**}=\lim T_n^{**}x^{**}$ is well defined 
for each $x^{**}\in X^{**}$. As 
$\{T_n\}$ is an \as/, $P$ is a projection onto $X$.
Let us show that $\ker P$ is $w^*$-closed. 
Indeed, if $x^{**}\in\ker P$, then 
$$
\|T_n^{**}x^{**}\|=\lim_m\|T^{\vphantom{**}}_mT_n^{**}x^{**}\|=
\lim_m\|T^{\vphantom{**}}_nT_m^{**}x^{**}\|=0
$$
and $T_n^{**}x^{**}=0$. Thus 
$$
\ker P=\bigcap_n\ker T_n^{**}.
$$
Let $M=(\ker P)_\perp$. Then $M^*=X$.\eck
%
\begin{corsub}
%
  Let \DE/.
  
  \bm{i} If $\SLE{1}$
\index{unconditional approximation property!for spaces $\SLE{1}$} 
has \uap/, then $E$ is a Riesz\index{Riesz set}
  set.
  
  \bm{ii} If $\SCE$
\index{unconditional approximation property!for spaces $\SCE$} 
has \uap/ and $\SCE\not\supseteq c_0$, then $E$
  is a Rosenthal set\index{Rosenthal set}.
%
\end{corsub}
%
\dem
In both cases, Lemma \ref{uap:sep} shows that 
the two spaces are separable dual spaces and thus have the 
Radon--Nikodym property. We may now apply 
Lust-Piquard's\index{Lust-Piquard, Fran\c coise} characterization
\cite{lu76}.\eck\vskipb 

There are Riesz sets $E$ such that $\SLE{1}$ fails \uap/: indeed, the
family of Riesz sets is coanalytic \cite{ta88}
%[Tardivel\index{Tardivel, Val\'erie}] 
while the second condition
is in fact analytic. There are Rosenthal sets that cannot be
sup-norm-partitioned\index{sup-norm-partitioned sets} \cite{bl74}.

The converse of Proposition \ref{sbd:ssesp}$(iii)$ does not hold: 
$\SLE{1}$ may have \uap/ while $\SCE$ fails this property. We have 
%
\begin{prpsub}\label{uap:uncond}
%
  Let \DE/.
  
  \bm{i} The Hardy space $H^1(\T)=\SL{1}{\sN}$ has \uap/.
  
  \bm{ii} The disc algebra $A(\T)=\SC{\sN}$ fails \uap/. More
  generally, if $\Z\setminus E$ is a Riesz set, then $\SCE$ fails
  \uap/.
%
\end{prpsub}
%
\dem
$(i)$
Indeed, $H^1(\T)$ has an unconditional basis \cite{ma80}. Note 
that the first unconditional \as/ for $H^1(\T)$ appears in 
\cite[\S II, introduction]{me68} with the help of 
Stein's\index{Stein, Elias} \cite{st66a,st66b} 
multiplier theorem (see also \cite{wo84}).

$(ii)$ 
Let $\Delta\subset\T$ be the Cantor set. By 
Bishop's\index{Bishop, Errett A.} 
improvement \cite{bi62} of 
Rudin\index{Rudin, Walter}--Carleson's\index{Carleson, Lennart} 
interpolation theorem, every function in 
$\script{C}(\Delta)$ extends to a function in $\SCE$ if 
$\Z\setminus E$ is a Riesz set. By 
\cite[main theorem]{pe64}, this implies that $\script{C}(\Delta)$ 
embeds in $\SCE$. Then $\SCE$ cannot have \uap/; otherwise
$\script{C}(\Delta)$ would 
embed in a space with an unconditional basis, which is false.\eck\vskipb 

\remsub
Recent studies of the Daugavet\index{Daugavet property}
Property by Kadets\index{Kadets, Vladimir M.}
and Werner\index{Werner, Dirk} 
generalize Proposition \ref{uap:uncond}$(ii)$. This property of a
Banach space $X$ states that for
every finite rank operator $T$ on $X$ $\|\Id+T\|=1+\|T\|$. 
By \cite[Th.\ 2.1]{ka96}, such an $X$ cannot have \uap/. Further, by
\cite[Th.\ 3.7]{we97}, $\SCE$ has the Daugavet Property if 
$\Z\setminus E$ is a
so-called semi-Riesz set\index{semi-Riesz set}, that is if 
all measures with
Fourier spectrum in 
$\Z\setminus E$ are diffuse. 

\vskipa\questsub
%What is the relation between $E$ and property \uap/ in $\SCE$~? 
Is there some characterization of sets \DE/ such that $\SCE$ has
\uap/~? Only a few classes of such sets are known: Sidon sets and
sup-norm-partitioned sets, for which $\SCE$ even has \lap{1}; certain
Hilbert sets. We conjecture that $\SCE$ fails \uap/ if $E$ contains 
an infinite parallelepiped.



\subsection{Characterization of \umap/ and \lpmap/}












Let us introduce 
%
\begin{dfnsub}\label{block:block:def}
%
  Let \DE/ and $X$ be a homogeneous Banach space on \T.
  
  $E$ enjoys the Fourier block\index{Fourier block unconditionality}
  unconditionality property \UP/ in $X$ whenever, for any $\eps>0$ and
  finite $F\se E$, there is a finite $G\se E$ such that for $f\in
  B_{X_F}$ and $g\in B_{X_{E\setminus G}}$
%
\begin{equation}\label{block:bloc}
%
\osc_{\epsilon\in\sU}\|\epsilon f+g\|_X\le\eps.
%
\end{equation}
%
\end{dfnsub}
%
\begin{lemsub}\label{blockapp:lem}
%
  Let \DE/ and $X$ be a homogeneous Banach space on \T. The following
  are equivalent.
  
  \bm{i} $X_E$ has $(u(\tau_f))$, where $\tau_f$ is the topology of
  pointwise convergence of the Fourier coefficients:
  $$
  x_n\hoch{\to}^{\tau_f}0\quad\Longleftrightarrow\quad \forall k\ 
  \widehat{x_n}(k)\to0.
  $$
  
  \bm{ii} $E$ enjoys \UP/ in $X$.
 
  \bm{iii} $X_E$ enjoys the property of block unconditionality for
  any, or equivalently for some, \as/ of multipliers $\{T_k\}$.
%
\end{lemsub}
%
\dem
%
$(i)\imp(ii)$. 
Suppose that $(ii)$ fails: there are 
$\eps>0$ and a finite $F$ such that for each finite $G$, there 
are $x_G\in B_{X_F}$
and $y_G\in B_{X_{E\setminus G}}$ 
such that 
$$
\osc_{\epsilon\in\sU}\|\epsilon x_G+y_G\|>\eps.
$$
As $B_{X_F}$ is compact, we may suppose $x_G=x$. 
As $y_G\hoch{\to}^{\tau_f}0$, $(u(\tau_f))$ fails. 

$(ii)\imp(iii)$. 
Let $C$ be a uniform bound for $\|T_k\|$. 
Let $n\ge1$ and $\eps>0$. Let $F$ be the finite spectrum of 
$T_n$. Let $G$ be such that $(\ref{block:bloc})$ holds 
for all $f\in B_{X_F}$ 
and $g\in B_{X_{E\setminus G}}$. 
Now there is a term $V$ in de la Vall\'ee-Poussin's 
\as/ such that $V|_{X_G}=\Id|_{X_G}$ and 
$\|V\|_{\script{L}(X_E)}\le3$. 
As $V$ has finite rank, we may choose $m>n$ such that 
$\|(\Id -T_m)V\|_{\script{L}(X_E)}=\|V(\Id -T_m)\|_{\script{L}(X_E)}\le\eps$. 
Let then $x\in T_nB_{X_E}$ and $y\in(\Id -T_m)B_{X_E}$. 
We have 
%
\begin{eqnarray*}
%
\|\epsilon x+y\|&\le&\|\epsilon x+(\Id -V)y\|+\eps
\hoch{\le}^{(\ref{block:bloc})}\|x+(\Id -V)y\|+4(C+1)\eps+\eps\\
&\le&\|x+y\|+(4C+6)\eps.%\\
%
\end{eqnarray*}

$(iii)\imp(i)$ is proved as Lemma \ref{block:lm} 
$(ii)\imp(i)$: note that if $y_j\hoch{\to}^{\tau_f}0$, 
then $\|Ty_j\|\to0$ for any finite rank multiplier $T$.
%
\eck
%
\vskipa 
We may now prove the main result of this section.
%
\begin{thmsub}\label{sbd:thm}
\index{metric unconditional approximation property!for homogeneous Banach spaces}
%
  Let \DE/ and $X$ be a homogeneous Banach space on \T.  If $X_E$ has
  \umap/, then $E$ enjoys \UP/ in $X$. Conversely, if $E$ enjoys \UP/
  in $X$ and furthermore $X_E$ has \uap/ and finite cotype, or simply
  \lap{1}, then $X_E$ has \umap/. In particular,
  
  \bm{i} For $1<p<\infty$, $\SLPE$ has \umap/ if and only if $E$
  enjoys \UP/ in $\SLP{}$.
  
  \bm{ii} $\SLE{1}$ has \umap/ if and only if $E$ enjoys \UP/ in
  $\SL{1}{}$ and $\SLE{1}$ has \uap/.
  
  \bm{iii} If $E$ enjoys \UP/ in $\SC{}$ and $\SCE$ has \lap{1}, in
  particular if $E$ is a Sidon set, then $\SCE$ has \umap/.
%
\end{thmsub}
%
\dem
%
Notice first that \umap/ implies \UP/ by Lemma \ref{blockapp:lem} 
$(iii)\imp(ii)$. 

$(i)$ 
Notice that $\SLPE$ $(1< p<\infty)$ has an unconditional \fdd/ of 
multipliers $\{\pi_{E\cap I_k}\}$ and cotype $2\vee p$. Thus \UP/
implies 
\umap/ by Theorem \ref{block:thm'}$(ii)\imp(iv)$. 

By Lemma \ref{blockapp:lem}, 
part $(ii)$ and $(iii)$ follow from 
Theorem \ref{block:thm}$(ii)\imp(iv)$ and 
Theorem \ref{block:thm'}$(ii)\imp(iv)$ respectively.
%
\eck\vskipb

\remsub\label{rem624}
Consider the special case $E=\{0\}\cup\{j^k\}_{k\ge0}$, $|j|\ge2$, and
suppose $X_E$ has complex \umap/. By Theorem \ref{sbd:thm},
$$
\osc_{\epsilon\in\sT}\|\epsilon a+b\e_{j^k}+c\e_{j^{k+1}}\|
\tol_{k\to\infty}0.
$$ 
Let us show that then $\{0,1,j\}$ is a
$1$-unconditional basic sequence in $X$. Indeed, for any 
$\epsilon,\mu,\nu\in\T$, and choosing $\kappa$ such that
$\mu\kappa=\nu\kappa^j$, 
%
\begin{eqnarray*}
%
%\lefteqn{\osc_{\epsilon,\mu,\nu\in\sT}\|\epsilon a+\mu b\e_1+\nu c\e_j\|_X=
%\osc_{\epsilon,\mu,\nu\in\sT}\|\epsilon a+\mu\nu
%b\e_1+\mu\nu^jc\e_j\|_X}\\
%&=&\osc_{\epsilon\in\sT}\|\epsilon a+b\e_1+c\e_j\|_X=
%\osc_{\epsilon\in\sT}\|\epsilon a+b\e_{j^k}+c\e_{j^{k+1}}\|_X\to 0.
%
%\lefteqn{
\|\epsilon a+\mu b\e_1+\nu c\e_j\|&=&
\|\epsilon a+\mu\kappa b\e_1+\nu\kappa^jc\e_j\|\\&=&
\|\epsilon\overline{\mu\kappa}a+b\e_1+c\e_j\|=
\|\epsilon\overline{\mu\kappa}a+b\e_{j^k}+c\e_{j^{k+1}}\|
%
\end{eqnarray*}
%
whose oscillation tends to $0$ with $k$. By Proposition
\ref{mub:isom}$(i)$, $X_E$ fails complex \umap/ if $X$ 
is $\SLP{}$, $p$ not an even integer, or $\SC{}$. By Proposition
\ref{mub:isom}$(ii)$, $\SLE{2n}$, $n\ge1$ integer, fails complex 
\umap/ if $j$ is positive and $n\ge j$, 
or if $j$ is negative and $n\ge|j|+1$.\vskipb

The study of \lpmap/ in $X_E$ reduces to the trivial case $p=2$,
$X=\SL{2}{}$, and to the case $p=1$, $X=\SC{}$. To see this, note that we have
by a repetition of the arguments of Lemma \ref{blockapp:lem} 
%
\begin{lemsub}\label{sbd:lpmap:lem}
%
  Let \DE/ and $X$ be a homogeneous Banach space.  The following
  properties are equivalent.
  
  \bm{i} $X_E$ has $m_p(\tau_f)$.
  
  \bm{ii} $E$ enjoys the following property $\script{M}_p$ in $X$: for
  any $\eps>0$ and finite $F\se E$, there is a finite $G\se F$ such
  that for $f\in B_{X_F}$ and $g\in B_{X_{E\setminus G}}$
  $$
  \bigl|\|f+g\|_X-(\|f\|_X^p+\|g\|_X^p)^{1/p}\bigr|\le\eps
  $$
  
  \bm{iii} $X_E$ enjoys $m_p(T_k)$ for any, or equivalently for some,
  \as/ of multipliers.
%
\end{lemsub}
%
\begin{prpsub}
\index{metric $p$-additive approximation property!for homogeneous Banach spaces}
%
  Let \DE/ and $X$ be a homogeneous Banach space.
  
  \bm{i} If $X_E$ has \lpmap/, then $E$ enjoys $\script{M}_p$ in $X$.
  
  \bm{ii} $\SLE{q}$ has \lpmap/ if and only if $p=q=2$.
  
  \bm{iii} 
\index{metric $1$-additive approximation property!for spaces $\SCE$} 
$\SCE$ has \lmap{1} if and only if it has \lap{1}
  and $E$ enjoys $\script{M}_1$ in $\SC{}$: for all $\eps>0$ and
  finite $F\se E$, there is a finite $G\se E$ such that
  $$
  \forall f\in\SC{F}\ \forall g\in\SC{E\setminus G}\qquad
  \|f\|_\infty+\|g\|_\infty\le(1+\eps)\|f+g\|_\infty.
  $$
%
\end{prpsub}
%
\dem 
$(i)$ 
Let $\eps>0$. Let $\{T_k\}$ be
an a.s.\ of multipliers that satisfies \Ref{block:def:str} with
$C<1+\eps$. By the
argument of \cite[Lemma 5]{li96}, we may assume that the $T_k$'s have 
their range in \PTE/.
Let $n\ge1$ be such that $\bigl(\sum_{k>n}\|\Delta
T_kf\|_X^p\bigr)^{1/p}<\eps$ for $f\in B_{X_F}$. Let $G$ be such that
$T_kg=0$ for $k\le n$ and $g\in X_{E\setminus G}$. Then successively 
$$
\Bigl|\Bigl(
\sum_{k\le n}\|\Delta T_k(f+g)\|_X^p\Bigr)^{1/p}-
\Bigl(\sum\|\Delta T_kf\|_X^p\Bigr)^{1/p}\Bigr|
\le\eps,
$$
$$
\Bigl|\Bigl(
\sum_{k> n}\|\Delta T_k(f+g)\|_X^p
\Bigr)^{1/p}-
\Bigl(\sum\|\Delta T_kg\|_X^p\Bigr)^{1/p}\Bigr|
\le\eps,
$$
$$
\Bigl|\Bigl(\sum
\|\Delta T_k(f+g)\|_X^p\Bigr)^{1/p}-
\Bigl(\sum\|\Delta T_kf\|_X^p+\sum\|\Delta T_kg\|_X^p\Bigr)^{1/p}
\Bigr|\le2^{1/p}\eps
$$
and 
$$
\bigl|
\|f+g\|_X-(\|f\|_X^p+\|g\|_X^p)^{1/p}
\bigr|\le2\eps(1+2^{1/p}).
$$
$(ii)$ 
By Corollary \ref{appen:uap}, we necessarily have $p=2$. Furthermore,
if $\SLE{q}$ has \lmap{2}, then by property $\script{M}_2$
$$
\bigl|\|\e_n+\e_m\|_q-\sqrt{2}\bigr|\tol_{m\to\infty}0.
$$
Now
$\|\e_n+\e_m\|_q=\|1+\e_1\|_q$ 
is constant and differs from $\|1+\e_1\|_2=\sqrt{2}$ unless $q=2$:
otherwise the only case of equality of the norms $\|\cdot\|_q$ and
$\|\cdot\|_2$ occurs for almost everywhere constant functions.

$(iii)$
Use Theorem \ref{block:strong:thm'}.
\eck











\section{Property \umap/ and arithmetical block independence}
\label{sect:umap}













We may now apply the technique used in the 
investigation of \umbs/ in order to obtain 
arithmetical conditions analogous to \I{n} (see Def.\ \ref{mub:def:ar})
for \umap/. According to Theorem \ref{sbd:thm}, it suffices 
to investigate property \UP/ of block unconditionality: we have 
to compute an expression of type $\|f+\epsilon g\|_p$, where the spectra 
of $f$ and $g$ are far apart and $\epsilon\in\U$. As before, $\U=\T$ 
(\vs $\U=\D$) is the complex (\vs real) choice of signs. 







\subsection{Property of block independence}\label{ss:block}







To this end, 
we return to the notation of Computational lemmas 
\ref{mub:calcul} and \ref{mub:culcul}. Define 
%
\begin{eqnarray}
%
\lefteqn{\nonumber\Psi_r(\epsilon,z)\quad=\quad\Theta_r((\overbrace{1,\dots,1}^j,
\overbrace{\epsilon,\dots,\epsilon}^{m-j}), z)}&&\\
\nonumber&=&\!\int
\biggl|\e_{r_0}(t)+
\sum_{i=1}^jz_i\e_{r_i}(t)
+\epsilon\sum_{i=j+1}^mz_i\e_{r_i}(t)\biggr|^pdm(t)\\
&=&\sum_{\setbox0\hbox{$\scriptstyle\alpha\in\sN^{m}$}\wd0=0pt\box0}\,\,
{\bip{p/2}\alpha}^2\prod|z_i|^{2\alpha_i}
\label{umap:thet}+
\sum_{\scriptstyle\alpha\ne\beta\in\sN^m\atop\scriptstyle\alpha\sim\beta}
%\sum_{A\in\script{R}_q}\sum_{\alpha\ne\beta\in A}
\bip{p/2}\alpha\bip{p/2}\beta
\epsilon^{\mathop{\lower1pt\hbox{$\Sigma$}}_{i>j}\!\alpha_i-\beta_i}
\prod z_i^{\alpha_i}
\overline{z_i}^{\beta_i}.
%
\end{eqnarray}
%
As in Computational lemma \ref{mub:culcul}, 
we make the following observation:
%
\begin{ldcsub}\label{umap:calcul}
%
  Let $\lstp\zeta0m\in\Z^*$ and $\gamma,\delta$ be as in
  $(\ref{mub:gammadelta})$. If the arithmetic relation
  $(\ref{mub:ArithRel})$ holds, then the coefficient of the term
  $\epsilon^{\lower1pt\hbox{$\Sigma$}_{i>j}\gamma_i-\delta_i} \prod
  z_i^{\gamma_i}\overline{z_i}^{\delta_i}$ in $(\ref{umap:thet})$ is
  $\bip{p/2}\gamma\bip{p/2}\delta$ and thus independent of $r$. If
  $\sum|\zeta_i|\le p$ or $p$ is not an even integer, this coefficient
  is nonzero. If $\smp\zeta0j$ is nonzero \(\vs odd\), then this term is
  nonconstant in $\epsilon\in\U$.
%
\end{ldcsub}
%
Thus the following arithmetical property shows up. It is 
similar to property \I{n}
of almost independence.
%
\begin{dfnsub}\label{arith:def}
%
  Let \DE/ and $n\ge1$.
  
  \bm{i} $E$ enjoys the complex \(\vs real\) property \J{n} of block
  independence\index{block independent set of integers} if for any
  $\zeta\in\Zeta_n^m$ with $\sm\zeta j$ nonzero \(\vs odd\) and given
  $\lst pj\in E$, there is a finite $G\se E$ such that
  $\zeta_1p_1+\dots+\zeta_mp_m\ne0$ for all $\lstp p{j+1}m\in
  E\setminus G$.
  
  \bm{ii} $E$ enjoys exactly complex \(\vs real\) \J{n} if furthermore
  it fails complex \(\vs real\) \J{n+1}.
  
  \bm{iii} $E$ enjoys complex \(\vs real\) \J{\infty} if it enjoys
  complex \(\vs real\) \J{n} for all $n\ge1$.
%
\end{dfnsub}
%
The complex (\vs real) property \J{n} means precisely the
following. ``For every finite $F\se E$ there is a finite $G\se E$ such
that for any two representations of any $k\in\Z$ as a sum of $n$
elements of $F\cup(E\setminus G)$ 
$$
\sm pn=k=\sm{p'}n
$$
one necessarily has
$$
\mes{\{j:p_j\in F\}}=\mes{\{j:p'_j\in F\}}\hbox{ in $\Z$ (\vs in $\Z/2\Z$).''}
$$
Thus property \J{n} has, unlike \I{n}, 
a complex and a real\index{real vs.\ complex}\index{complex vs.\ real} version. Real \J{n} is strictly 
weaker than complex \J{n}: see Section \ref{sect:arith}. 
Notice that \J{1} is void and $\J{n+1}\imp\J{n}$ in both 
complex and real cases. Also $\I{n}\not\imp\J{n}$: 
we shall see in the following section that $E=\{0\}\cup\{n^k\}_{k\ge0}$ 
provides a counterexample. 
The property \J{2} of 
real block independence appears implicitly in 
\cite[Lemma 12]{li96}.\vskipb

\remsub 
In spite of the intricate form of this 
arithmetical property, \J{n} is the ``simplest'' candidate, 
in some sense, that reflects the features of \UP/: 

\bloc 
it must hold for a set $E$ if and only if it holds 
for a translate $E+k$ of this set: this explains 
$\sum\zeta_i=0$ in Definition \ref{arith:def}$(i)$; 

\bloc 
as for the property \UP/ of block independence, it must 
connect the break\index{break} of $E$ with its tail\index{tail}; 

\bloc 
Li gives an example of a set $E$ whose pace does not 
tend to infinity while $\SCE$ has \lmap{1}. 
Thus no property \J{n} should forbid parallelogram 
relations of the type $p_2-p_1=p_4-p_3$, where 
$p_1,p_2$ are in the break of $E$ and 
$p_3,p_4$ in its tail. This 
explains the condition that $\sm\zeta j$ be nonzero 
(\vs odd) in 
Definition \ref{arith:def}$(i)$.\vskipb

We now repeat the argument of Theorem \ref{mub:thm}
to obtain an analogous statement which relates 
property \UP/ of Definition \ref{block:block:def} 
with our new arithmetical conditions
%
\begin{lemsub}\label{umap:lem}
  \index{Fourier block unconditionality} 
  \index{block independent set of integers}
%
  Let \DEE/ and $1\le p<\infty$.
  
  \bm{i} Suppose $p$ is an even integer. Then $E$ enjoys the complex
  \(\vs real\) Fourier block unconditionality property \UP/ in
  $\SLP{}$ if and only if $E$ enjoys complex \(\vs real\) \J{p/2}.
  
  \bm{ii} If $p$ is not an even integer and $E$ enjoys complex \(\vs
  real\) \UP/ in $\SLP{}$, then $E$ enjoys complex \(\vs real\)
  \J{\infty}.
%
\end{lemsub}
%
\dem
Let us first prove the necessity of the arithmetical 
property and assume $E$ fails \J{n}: then 
there are $\lstp\zeta0m\in\Z^*$ with $\sum\zeta_i=0$, 
$\sum|\zeta_i|\le2n$ and $\smp\zeta0j$ nonzero (\vs odd); 
there are $\lstp{r}0j\in E$ and sequences 
$\lstp{r^l}{j+1}m\in E\setminus\{\lstp n1l\}$ such that
$$
\zeta_0r_0+\dots+\zeta_jr_j+\zeta_{j+1}r_{j+1}^l+\dots+\zeta_mr_m^l=0.
$$ 

Assume $E$ enjoys \UP/ in $\SLP{}$. Then
the oscillation of $\Psi_r$ in $(\ref{umap:thet})$ satisfies
%
\begin{equation}\label{umap:ApplDef}
%
\osc_{\epsilon\in\sU}
\Psi_{r^l}(\epsilon,z)
\tol_{l\to\infty}0
%
\end{equation}
%
for each $z\in D^m$. The argument is now 
exactly the same as in Theorem \ref{mub:thm}: 
we may assume that 
the sequence of functions $\Psi_{r^l}$ converges 
in $\script{C}^\infty(\U\times D^m)$ to a function 
$\Psi$. Then by $(\ref{umap:ApplDef})$,
$\Psi(\epsilon,z)$ is constant in $\epsilon$ 
for each $z\in D^m$, and this is impossible by 
Computational lemma \ref{umap:calcul} 
if $p$ is either not an even integer or
$p\ge 2n$.

Let us now prove the sufficiency of \J{p/2} when 
$p$ is an even integer.
First, let $\Alpha_n^{k,l}=
\{\alpha\in \Alpha_n:\alpha_i=0\mbox{ for }k<i\le l\}$ 
($\Alpha_n$ is defined before Prop.\ \ref{mub:isom}), and
convince yourself that \J{p/2} is equivalent to
%
\begin{equation}\forall k\ \exists l\ge k\ 
\forall\alpha,\beta\in \Alpha_{p/2}^{k,l}\quad 
\sum\alpha_in_i=\sum\beta_in_i\ 
\label{umap:DefEq}\imp\ \sum_{i\le k}\alpha_i=
\sum_{i\le k}\beta_i\ (\mbox{\vs mod $2$}).
\end{equation}
%
Let $f=\sum a_i\e_{n_i}\in\PTE/$. Let $k\ge1$ and 
$\epsilon\in\U$. By the multinomial formula, 
\begin{eqnarray*}
\lefteqn{\|\epsilon \pi_kf+(f-\pi_{l}f)\|_p^p=
\int{\biggl|\sum_{\alpha\in \Alpha_{p/2}^{k,l}}
\bip {p/2}\alpha\epsilon^
{\lower1pt\hbox{$\Sigma$}_{p\le k}\alpha_i}
\biggl(\prod a_i^{\alpha_i}
\biggr)\e_{\lower1pt\hbox{$\Sigma$}\alpha_in_i}\biggr|}^2dm}\qquad\qquad\\
&&=\int{\biggl|\sum_{j=0}^n\epsilon^j
\sum_
{\scriptstyle\alpha\in \Alpha_{p/2}^{k,l}\atop
 \scriptstyle\sm\alpha k=j}
\bip{p/2}\alpha\biggl(\prod a_i^{\alpha_i}
\biggr)\e_{\lower1pt\hbox{$\Sigma$}\alpha_in_i}\biggr|}^2dm.
\end{eqnarray*}
$(\ref{umap:DefEq})$ now signifies that we may choose 
$l\ge k$ such that the terms of the above sum over $j$ 
(\vs the terms with $j$ odd 
and those with $j$ even) have 
disjoint spectrum. But then 
$\|\epsilon \pi_kf+(f-\pi_{l}f)\|_p$ is constant for 
$\epsilon\in\U$ and $E$ enjoys \UP/ in $\SLP{}$.
\eck\vskipb

Note that for even $p$, we have as in 
Proposition \ref{mub:isom} a constant $C_p>1$ such that either
$(\ref{block:bloc})$ holds for $\eps=0$ or fails 
for any $\eps\le C_p$. We thus get
%
\begin{corsub}
%
  Let \DE/ and $p$ be an even integer. If $E$ enjoys complex \(\vs
  real\) \UP/ in $\SLP{}$, then there is a partition $E=\bigcup E_k$ into
  finite sets such that for any coarser partition $E=\bigcup E'_k$
  $$
  \forall f\in\PTE/ \quad\osc_{\epsilon_k\in\sU}\Bigl\|\sum
  \epsilon_k\pi_{E'_{2k}}f\Bigr\|_p=0
  $$
%
\end{corsub}
%
Among other consequences, $E=E_1\cup E_2$ 
where the $\SLP{E_i}$ have a complex (\vs real) 
$1$-unconditional 
\fdd/\index{$1$-unconditional fdd@$1$-unconditional \fdd/}.\vskipb

\questsub
Is this rigidity proper to translation invariant subspaces of $\SLP{}$ with
$p$ an even integer, or generic for all its subspaces 
(see 
%recent work by Delbaen--Jarchow--Pe\l czy\'nski 
\cite{djp})~?












\subsection{Main result}




Lemma \ref{umap:lem} and Theorem \ref{sbd:thm} yield the 
main result of this section.
%
\begin{thmsub}\label{umap:thm}
\index{metric unconditional approximation property!for spaces $\SLPE$, $p$ even}
%
  Let \DE/ and $1\le p<\infty$.
  
  \bm{i} Suppose $p$ is an even integer. Then $\SLPE$ has complex
  \(\vs real\) \umap/ if and only if $E$ enjoys complex \(\vs real\)
  \J{p/2}.
  
  \bm{ii} If $p$ is not an even integer and $\SLPE$ has complex \(\vs
  real\) \umap/, then $E$ enjoys complex \(\vs real\) \J{\infty}.
%
\end{thmsub}
\begin{corsub}\label{umap:thm:cor}\index{metric unconditional approximation property!for $\SCE$ and $\SLPE$, $p\notin2\N$}

%
  Let \DE/.
  
  \bm{i} If $\SCE$ has complex \(\vs real\) \umap/, then $E$ enjoys
  complex \(\vs real\) \J{\infty}.
  
  \bm{ii} If any $\SLPE$, $p$ not an even integer, has complex \(\vs
  real\) \umap/, then all $\SLPE$ with $p$ an even integer have
  complex \(\vs real\) \umap/.
%
\end{corsub}
Suppose $p$ is an even integer. Then Section \ref{sect:arith} 
gives various examples of sets such that $\SLPE$ 
has complex or real \umap/. Proposition \ref{comb:grow}
gives a general growth condition that ensures \umap/.

For $X=\SLP{}$, $p$ not an even integer, 
and $X=\SC{}$, however, we encounter the same obstacle 
as for \umbs/. Section \ref{sect:arith} only gives 
sets $E$ such that $X_E$ fails \umap/. Thus, we 
have to prove this property by direct means. This yields four 
types of examples of sets $E$ such that the space 
$\SCE$ --- and thus by \cite[Th.\ 7]{li96} all 
$\SLPE$ $(1\le p<\infty)$ as well --- have \umap/. 

\bloc Sets found by Li\index{Li, Daniel} \cite{li96}: 
Kronecker's
%\cite{Kronecker, Leopold} 
theorem is used 
to construct a set containing arbitrarily long 
arithmetic sequences and a set whose pace does not 
tend to infinity. Meyer's\index{Meyer, Yves} \cite[VIII]{me72} techniques are 
used to construct a Hilbert set\index{Hilbert set}.

\bloc 
The sets that satisfy the growth condition of Theorem
\ref{positif:thm};

\bloc
Sequences \DEE/ such that $n_{k+1}/n_k$ is an odd integer: 
see Proposition \ref{res:geo}.\vskipb

\questsub 
We know no example of a set $E$ such that 
some $\SLPE$, $p$ not an even integer, has \umap/ 
while $\SCE$ fails it.\vskipb

There is also a good arithmetical description of the case 
where $\{\pi_k\}$ or a subsequence 
thereof realizes \umap/. 
%
\begin{prpsub}\label{umap:prp:fdd}
%
\index{$1$-unconditional fdd@$1$-unconditional \fdd/!for spaces $\SLPE$, $p$ even} 
\index{metric unconditional fdd@metric unconditional \fdd/} 
Let \DEEE/. Consider a partition
  $E=\bigcup_{k\ge1}E_k$ into finite sets.
  
  \bm{i} Suppose $p$ is an even integer. The series $\sum\pi_{E_k}$
  realizes complex \(\vs real\) \umap/ in $\SLPE$ if and only if there
  is an $l\ge1$ such that
\begin{equation}
\label{umap:fdd}
\left\{\begin{array}{l}
\lst pm\in E\\
\zeta_1p_1+\dots+\zeta_mp_m=0
\end{array}\right.
\quad\imp\quad
\forall k\ge l\suml_{p_j\in E_k}\zeta_j=0
\ (\mbox{\vs is even})
\end{equation}
for all $\zeta\in\Zeta_{p/2}^m$.  Then $\SLPE$ admits the series
$\pi_{\cup_{k<l}E_k}+\sum_{k\ge l}\pi_{E_k}$ as
$1$-un\-con\-di\-tio\-nal \fdd/. In particular, choose $E_k=\{n_k\}$.
The sequence $\{\pi_k\}$ realizes complex and real \umap/ in $\SLPE$
if and only if there is a finite $G$ such that for
$\zeta\in\Zeta_{p/2}^m$
%
\begin{equation}\label{umap:base}
%
\left\{\begin{array}{l}
\lst pm\in E\\
\zeta_1p_1+\dots+\zeta_mp_m=0
\end{array}\right.
\quad\imp\quad\lst pm\in G.
%
\end{equation}
%
Then $E\setminus G$ is a $1$-\ubs/ and $E$ enjoys \I{p/2}.

\bm{ii} Suppose $p$ is not an even integer. If $\sum\pi_{E_k}$
realizes complex \(\vs real\) \umap/ in $\SLPE$, then for each
$\zeta\in\Zeta^m$ there is an $l\ge1$ such that $(\ref{umap:fdd})$
holds.  In particular, if $\{\pi_k\}$ realizes either complex or real
\umap/ in $\SLPE$, then for all $\zeta\in\Zeta^m$ there is a finite
$G$ such that $(\ref{umap:base})$ holds. This is equivalent to \I{\infty}. 
%
\end{prpsub}
%
\dem 
It is analogous to the proof of Lemma \ref{umap:lem}: 
suppose we have $\zeta\in\Zeta_n^m$ such that 
$(\ref{umap:fdd})$ fails for any $l\ge1$. 
Then there are $\lstp\zeta0m\in\Z^*$ with 
$\sum\zeta_i=0$, $\sum|\zeta_i|\le2n$ and 
$\smp\zeta0j$ nonzero (\vs odd) for some $j$; 
for each $l$, 
there are $\lstp{r^l}0j\in\cup_{k<l}E_k$ and 
$\lstp{r^l}{j+1}m\in\cup_{k\ge l}E_k$
such that $\zeta_0r_0^l+\dots+\zeta_mr_m^l=0$. 

But then $\sum\pi_{E_k}$ cannot 
realize complex (\vs real) \umap/: the function 
$\Psi_r$ in $(\ref{umap:thet})$ would satisfy 
$(\ref{umap:ApplDef})$ and we would 
obtain a contradiction 
as in Theorem \ref{mub:thm}.

Sufficiency in $(i)$ and $(i')$ is proved exactly as in 
Lemma \ref{umap:lem}$(i)$.
\eck\vskipb

In particular, suppose that the cardinal 
$\mes{E_k}$ is uniformly 
bounded by $M$ and $\{\pi_{E_k}\}$ realizes \umap/ in 
$\SLPE$. If $p\ne2$ is an even integer, then $E$ 
is a \EL{p}\index{Lambda(p) set@\EL{p} set} set as union of a 
finite set and $M$ $p/2$-independent sets 
(see Prop.\ \ref{mub:isom} and 
\cite[\protect{Th.\ 4.5(b)}]{ru60}). 
If $p$ is not an even 
integer, then $E$ is a \EL{q} set for all $q$ 
by the same argument. 

%Let $p$ be an even integer. 
%If $\{\pi_k\}$ realizes \umap/ in $\SLPE$, then $E$ is {\it a
% fortiori}\/ a \umbs/ in
%$\SLP{}$. 

%Let $p$ be an even integer. If $(\ref{umap:fdd})$ holds, then a tail
%of $\{U_k\}$ 
%realizes $1$-\fdd/ in $\SLPE$. Similarly, if 
%$(\ref{umap:base})$ holds, then $E$ is a \umbs/ 
%in $\SLP{}$.












\section{Examples for \umap/: block independent sets of characters}
\label{sect:arith}







\subsection{General properties}









The pairing $\XE$ underlines the asymptotic nature of property $\J{n}$. 
It has been defined before Proposition \ref{mub:lim}, whose proof adapts to 
%
\begin{prpsub}\label{arith:lim}
%
  Let \DEE/.
  
  \bm{i} If $\XE<\infty$ for $\lst\zeta m\in\Z^*$ with $\sum\zeta_i$
  nonzero \(\vs odd\), then $E$ fails complex \(\vs real\)
  \J{|\zeta_1|+\dots+|\zeta_m|}.  Conversely, if $E$ fails complex
  \(\vs real\) \J{n}, then there are $\lst\zeta m\in\Z^*$ with
  $\sum\zeta_i$ nonzero \(\vs odd\) and $\sum|\zeta_i|\le2n-1$ such
  that $\XE<\infty$.
  
  \bm{ii} Thus $E$ enjoys complex \(\vs real\) \J{\infty} if and only
  if $\XE=\infty$ for all $\lst\zeta m\in\Z^*$ with $\sum\zeta_i$
  nonzero \(\vs odd\).
%
\end{prpsub}
%
{\it Proof of the converse in $(i)$.\/} If $E$ fails complex (\vs real) 
\J{n}, then there are $\zeta\in\Zeta_n^m$ with 
$\sm\zeta j$ nonzero (\vs odd), $\lst pj\in E$ and 
sequences $\lstp{p^l}{j+1}m\in\{n_k\}_{k\ge l}$ 
such that 
$\sum_{i>j}\zeta_{i}p^l_i=
-\sum_{i\le j}\zeta_ip_i$. Let 
$\zeta'=(\zeta_{j+1},\dots,\zeta_m)$. Then 
$\sum|\zeta'_i|\le2n-1$ and 
$\langle\zeta',E\rangle<\infty$.\eck\vskipb

An immediate application is, as in Proposition \ref{mub:lim}, 
%
\begin{prpsub}\label{arith:csq}
%
  Let \DEE/.
  
  \bm{i} Suppose $E$ enjoys \I{2n-1}.  Then $E$ enjoys complex \J{n}
  and actually there is a finite set $G$ such that $(\ref{umap:base})$
  holds for $\zeta\in\Zeta_n^m$.
  
  \bm{ii} Suppose $E$ enjoys \I{\infty}. Then $E$ enjoys complex
  \J{\infty} and actually for all $\zeta\in\Zeta^m$ there is a finite
  $G$ such that $(\ref{umap:base})$ holds.
  
  \bm{iii} Complex and real \J{\infty} are stable under bounded
  perturbations of $E$.
  
  \bm{iv} Suppose there is $h\in\Z$ such that $E\cup\{h\}$ fails
  complex \(\vs real\) \J{n}. Then $E$ fails complex \(\vs real\)
  \J{2n-1}. Thus the complex and real properties \J{\infty} are stable
  under unions with an element: if $E$ enjoys it, then so does
  $E\cup\{h\}$.
  
  \bm{v} Suppose $jF+s,kF+t\in E$ for an infinite $F$, $j\ne k\in\Z^*$
  and $s,t\in\Z$. Then $E$ fails complex \J{|j|+|k|}, and also real
  \J{|j|+|k|} if $j$ and $k$ have different parity.
%
\end{prpsub}

We now turn to an arithmetical investigation of various sets $E$.







\subsection{Geometric sequences}\index{geometric sequences}








Let $G=\{j^k\}_{k\ge0}$ with 
$j\in\Z\setminus\{-1,0,1\}$. We resume Remark \ref{rem624}.

\vskipa{\bf (1) }
As $G,jG\se G$, $G$ 
fails complex \J{|j|+1}, and also real \J{|j|+1} if 
$j$ is even. The solutions $(\ref{arith:sol})$ 
to the Diophantine equation 
$(\ref{arith:dioph})$ show at once that $G$ enjoys complex
\J{|j|}, since there is no arithmetical relation
$\zeta\in\Zeta_{|j|}^m$ between the break and the tail
of $G$. If $j$ is odd, then $G$ enjoys in fact real \J{\infty}. 
Indeed, let $\lst\zeta m\in\Z^*$ and $k_1<\dots<k_m$: then 
$\sum\zeta_i j^{k_i}\in j^{k_1}\Z$ and either 
$\big|\sum\zeta_i j^{k_i}\big|\ge j^{k_1}$ or 
$\sum\zeta_i j^{k_i}=0$. 
Thus, if $\XE<\infty$ then $\XE=0$ and 
$\sum\zeta_i$ is even 
since $j$ is odd. Now apply Proposition \ref{arith:lim}$(iii)$. 
The same argument yields that even $G\cup-G\cup\{0\}$ 
enjoys real \J{\infty}. Actually much more is true: 
see Proposition \ref{res:geo}.

\vskipa{\bf (2) }
$G\cup\{0\}$ may behave differently 
than $G$ with respect to \J{n}: 
thus this property is not stable under unions with an 
element. Indeed, the first 
solution in $(\ref{arith:sol})$ may be written as 
$(-j+1)\cdot0+j\cdot j^k+(-1)\cdot j^{k+1}=0$. 
If $j$ is positive, $(-j+1)+j+(-1)\le2j$ and 
$G\cup\{0\}$ fails complex \J{j}. 
A look at $(\ref{arith:sol})$ shows that 
it nevertheless 
enjoys complex \J{j-1}. On the other hand, 
$G\cup\{0\}$ still enjoys complex 
\J{|j|} if $j$ is negative. 
In the real setting, our arguments yield 
the same if $j$ is even, but 
we already saw that $G\cup\{0\}$ 
still enjoys real \J{\infty} if $j$ is odd.







\subsection{Symmetric sets}\index{symmetric sets}







By Proposition \ref{mub:lim}$(iii)$ and \ref{arith:csq}$(vi)$, they do 
enjoy neither \I{2} nor complex \J{2}. They may 
nevertheless enjoy real \J{n}. Introduce property 
$(\script{J}^{\mbox{\tiny sym}}_n)$ for $E$: 
it holds if 
for all $\lst pj\in E$ and $\eta\in{\Z^*}^m$ with 
$\sum_1^m\eta_i$ even, $\sum_1^m|\eta_i|\le2n$ and 
$\sm\eta j$ odd, there is a finite set $G$ such that 
$\eta_1p_1+\dots+\eta_mp_m\ne0$ for 
any $\lstp p{j+1}m\in E\setminus G$. Then we obtain
%
\begin{prpsub}\label{arith:sym} 
%
  $E\cup-E$ has real \J{n} if and only if $E$ has
  $(\script{J}^{\mbox{\tiny sym}}_n)$.
%
\end{prpsub}
%
\dem
By definition, 
$E\cup-E$ has real \J{n} if and only if for all 
$\lst pj\in E$ 
and $\zeta^1,\zeta^2\in\Z^m$ with $\zeta^1+\zeta^2\in\Zeta_n^m$ 
and odd $\sum_{i\le k}\zeta^1_i-\zeta^2_i$, there is a 
finite set $G$ such that $\sum(\zeta^1_i-\zeta^2_i)p_i\ne0$ 
for any $\lstp p{j+1}m\in E\setminus G$ ---~and 
thus if and only if $E$ enjoys 
$(\script{J}^{\mbox{\tiny sym}}_n)$: 
just consider 
the mappings between arithmetical relations 
$(\zeta^1,\zeta^2)\mapsto\eta=\zeta^1-\zeta^2$
and 
$\eta\mapsto(\zeta^1,\zeta^2)$ such that
$\eta=\zeta^1-\zeta^2$, where
$\zeta^1_i=\eta_i/2$ if $\eta_i$ is even and,
noting that the number of odd $\eta_i$'s must be even, 
$\zeta^1_i=(\eta_i-1)/2$ and $\zeta^1_i=(\eta_i+1)/2$ 
respectively for each half of them.\eck

Consider again a geometric sequence $G=\{j^k\}$ with $j\ge2$. 
If $j$ is odd, we saw before that 
$G\cup-G$ and $G\cup-G\cup\{0\}$ enjoy real \J{\infty}.
If $j$ is even, then $G\cup-G$ fails 
real $\J{j+1}$ since $G$ does. 
$G\cup-G\cup\{0\}$ fails real \J{j/2+1}
by the arithmetical relation 
$1\cdot0+j\cdot j^k+(-1)\cdot j^{k+1}=0$
and Proposition \ref{arith:sym}. 
$G\cup-G$ enjoys real 
\J{j} and $G\cup-G\cup\{0\}$ enjoys real 
\J{j/2} as the 
solutions in $(\ref{arith:sol})$ show by a 
simple checking.







\subsection{Algebraic and transcendental numbers}
\index{transcendental numbers}






 
The proof of Proposition \ref{mub:trans} adapts to 
%
\begin{prpsub}\label{arith:trans}
%
  Let \DEE/.
  
  \bm{i} If $n_{k+1}/n_k\to\sigma$ where $\sigma>1$ is transcendental,
  then $E$ enjoys complex \J{\infty}.
  
  \bm{ii} Let $n_k=[\sigma^k]$ with $\sigma>1$ algebraic. Let
  $P(x)=\zeta_0+\dots+\zeta_dx^d$ be the corresponding polynomial of
  minimal degree. Then $E$ fails complex
  \J{|\zeta_0|+\dots+|\zeta_d|}, and also real
  \J{|\zeta_0|+\dots+|\zeta_d|} if $P(1)$ is odd.
%
\end{prpsub}






\subsection{Polynomial sequences}\index{polynomial sequences}







Let $E=\{P(k)\}$ for a polynomial $P$ of degree $d$. 
The arithmetical relation $(\ref{mub:poly})$ does 
not adapt to property \J{n}. Notice, though, that 
$\{\Delta^jP\}_{j=1}^d$ is a basis for the space 
of polynomials of degree less than $d$ and that 
$2^dP(k)-P(2k)$ is a polynomial of degree at most $d-1$. 
Writing it in the basis $\{\Delta^jP\}_1^d$ yields 
an arithmetical relation 
$2^d\cdot P(k)-1\cdot P(2k)+
\sum_{j=0}^d\zeta_j\cdot P(k-j)=0$ such that 
$2^d-1+\sum\zeta_j$ is odd. By Proposition \ref{arith:lim}
$(ii)$, 
$E$ fails 
real \J{n} for a certain $n$. This $n$ may be bounded in certain cases:

\bloc The set of squares\index{polynomial sequences!squares} fails real \J{2}:
let $F_n$ be the Fibonacci\index{Fibonacci sequence} sequence defined by $F_0=F_1=1$ 
and $F_{n+2}=F_{n+1}+F_n$. As 
$\{F_{n+1}/F_n\}$ is the sequence of convergents of 
the 
continued fraction associated to an irrational 
(the golden 
ratio), $F_n\to\infty$ and 
$F_nF_{n+2}-F_{n+1}^2=(-1)^n$ (see \cite{eu62}). 
Inspired by
\cite[p.\ 15]{mo69}, we observe that
$$
(F_nF_{n+2}+F_{n+1}^2)^2+
(F_{n+1}^2)^2=
(F_nF_{n+1}+F_{n+1}F_{n+2})^2+1^2
$$

\bloc
The set of cubes\index{polynomial sequences!cubes} fails real $\J{2}$:
starting from Binet's\index{Binet, J. P. M.} 
\cite{bi41} simplified solution 
of Euler's\index{Euler, Leonhard} equation \cite{eu56}, we observe that
$p_n=9n^4$, $q_n=1+9n^3$, $r_n=3n(1+3n^3)$ satisfy
$p_n^3+q_n^3=r_n^3+1^3$ and tend to infinity.

\bloc
The set of biquadrates\index{polynomial sequences!biquadrates} fails real $\J{3}$:
by an equality of Ramanujan\index{Ramanujan, Srinivasa} (see \cite[p.\ 386]{ra57}),
$$
(4n^5-5n)^4+(6n^4-3)^4+(4n^4+1)^4=
(4n^5+n)^4+(2n^4-1)^4+3^4.
$$

As for \I{n}, a positive answer to Euler's 
conjecture\index{Euler's conjecture} would imply that 
the set of $k$th powers has complex \J{2} for $k\ge5$. \vskipb

{\bf Conclusion } 
By Theorem \ref{umap:thm}, 
property \J{n} yields directly 
\umap/ in the space $\SL{2p}{}$, $p\le n$ integer. 
But we do not know whether
\J{\infty} ensures \umap/ in spaces 
$\SLP{}$, $p$ not an even integer, or $\SC{}$. 

Nevertheless, 
the study of property \J{3} permits us 
to determine the density of sets 
such that $X_E$ enjoys \umap/ for some $X\ne\SL{2}{}, \SL{4}{}$: see 
Proposition \ref{comb:thm}. Other 
applications are given in Section \ref{sect:resume}. 

















\section{Positive results: parity and a sufficient 
growth condition}
\label{sect:positif}










\subsection[$\SC{\{3^k\}}$ has real \umap/ because $3$ is odd]
{${\hbox{\zwe C}}_{\{3^k\}}({\hbox{\zwm T}})$ has real \umap/ 
because $3$ is odd}





In the real case, parity plays an unexpected r\^ole.
%
\begin{prpsub}\label{res:geo}\index{geometric sequences}
%
  Let \DEE/ and suppose that $n_{k+1}/n_k$ is an odd integer for all
  sufficiently large $k$.  Then $\SCE$ has real \umap/.
%
\end{prpsub}
%
Then $X_E$ also has real \umap/ for every homogeneous Banach space $X$
on $\T$.\vskipb

\dem
%
Let us verify that 
real \UP/ holds. Let $\eps>0$ and 
$F\se E\cap[-n,n]$. Let $l$, to 
be chosen later, such that 
$n_{k+1}/n_k$ is an odd 
integer for $k\ge l$. Take 
$G\supseteq\{\lst nl\}$ finite. 
Let $f\in B_{\script{C}_F}$ and 
$g\in B_{\script{C}_{E\setminus G}}$. Then 
$g(u\exp{\ii\pi/n_l})=-g(u)$ 
and 
$$
|f(u\exp{\ii\pi/n_l})-f(u)|
\le\pi/|n_l|\cdot\|f'\|_\infty\le
\pi n/|n_l|\le\eps
$$ 
by Bernstein's inequality and for $l$ large enough. 
Thus, for some $u\in\T$,
%
\begin{eqnarray*}
%
\|f-g\|_\infty&=&|f(u)+g(u\exp{\ii\pi/n_l})|\\
&\le&|f(u\exp{\ii\pi/n_l})+g(u\exp{\ii\pi/n_l})|+\eps\\
&\le&\|f+g\|_\infty+\eps.
%
\end{eqnarray*}
%
As $E$ is a Sidon set, 
we may apply Theorem \ref{sbd:thm}$(iii)$.
\eck\vskipb

Furthermore, if $E$ satisfies the hypothesis
of Proposition \ref{res:geo}, so does
$E\cup-E=\{n_1,-n_1,n_2,-n_2,\dots\}$\index{symmetric sets}.
But $E\cup-E$ fails even complex \J{2}
and no $X_{E\cup-E}\ne\SL{2}{E\cup-E}$ has complex \umap/.
On the other hand, if there is an even integer $h$ such that 
$n_{k+1}/n_k=h$ infinitely often, 
then $E$ fails real \J{|h|+1} by Proposition
\ref{arith:csq}$(vi)$.\vskipb 

\remsub
Note that if $n_{k+1}/n_k$ is furthermore uniformly bounded, then the
a.s.\ that realizes \umap/ cannot be too simple. In particular, it
cannot be a \fdd/ in translation invariant spaces $\SC{E_i}$: let $k$ be
such that $n_k$ and $n_{k+1}$ are in distinct $E_i$; then
$n_{k+1}+(-n_{k+1}/n_k)\cdot n_k=0$ and we may apply Proposition
\ref{umap:prp:fdd}$(ii)$. This justifies the use of Theorem
\ref{sbd:thm}$(iii)$. 








\subsection[Growth conditions: the case $\SLP{}$, $p$ an even integer]
{Growth conditions: the case ${\fam0 L}^p(\hbox{\zwm T})$, $p$
 an even integer}






For $X=\SLP{}$ with $p$ an even integer, 
a look at \I{n} and \J{n} gives by 
Theorems \ref{mub:thm} and \ref{umap:thm} 
the following general growth condition: 
%
\begin{prpsub}\label{comb:grow}
\index{$1$-unconditional approximation property!for spaces $\SLPE$, $p$ even} 
\index{$1$-unconditional basic sequence of characters!in spaces $\SLP{}$, $p$ even}
%
  Let \DEE/ and $p\ge1$ an integer. If
\begin{equation}\label{comb:grow:a}
\liminf|n_{k+1}/n_k|\ge p+1,
\end{equation} 
then $\{\pi_k\}$ realizes the complex \umap/ in $\SLE{2p}$ and there is
a finite $G\se E$ such that $E\setminus G$ is a $1$-unconditional
basic sequence in $\SL{2p}{}$.
%
\end{prpsub}
%
\dem
Suppose we have an arithmetical relation 
\begin{equation}\label{comb:eqn}
\zeta_1n_{k_1}+\dots+\zeta_mn_{k_m}=0\quad\mbox{with}\quad
\zeta\in\Zeta_p^m\mbox{ and }|n_{k_1}|<\dots<|n_{k_m}|.
\end{equation} 
Then 
$|\zeta_mn_{k_m}|\le|\zeta_1n_{k_1}|+\dots
+|\zeta_{m-1}n_{k_{m-1}}|$. 
The left hand 
side is smallest when $|\zeta_m|=1$. As 
$|\zeta_1|+\dots+|\zeta_m|\le 2p$ and necessarily 
$|\zeta_i|\le p$, the right hand side is largest 
when $|\zeta_{m-1}|=p$ and $|\zeta_{m-2}|=p-1$. 
Furthermore, it is largest when 
$k_m=k_{m-1}+1=k_{m-2}+2$. 
Thus, if $(\ref{comb:eqn})$ 
holds, then 
$$|n_{k_m}|\le p|n_{k_{m-1}}|+(p-1)|n_{k_{m-2}}|.$$
By $(\ref{comb:grow:a})$, this is impossible as soon 
as $m$ is chosen sufficiently large, because $p+1>p+(p-1)/(p+1)$.
%
\eck\vskipb

Note that Proposition \ref{comb:grow} is best 
possible: if $j$ is negative, then $\{j^k\}$ fails 
\I{|j|}. If $j$ is positive, 
then $\{j^k\}\cup\{0\}$ fails complex \J{j}. 







\subsection{A general growth condition}








Although we could prove that $E$ enjoys \I{\infty} and \J{\infty} when 
$n_{k+1}/n_k\to\infty$, we need a direct 
argument in order 
to get the corresponding functional properties: we have 
%
\begin{thmsub}\label{positif:thm}
%
  Let \DEE/ such that $n_{k+1}/n_k\to\infty$.  Then $\SCE$ has
\lmap{1}\index{metric $1$-additive approximation property!for spaces $\SCE$} 
with $\{\pi_k\}$ and $E$ is a 
Sidon\index{Sidon set!with constant asymptotically $1$} 
    set with constant
  asymptotically $1$.  If the ratios $n_{k+1}/n_k$ are all integers,
  then the converse holds.
%
\end{thmsub}
%
Note that by Proposition \ref{mub:pinf}$(ii)$, $E$ is a
metric unconditional basic sequence
\index{metric unconditional basic sequence} 
in every homogeneous Banach space
$X$ on $\T$. Further $X_E$ has complex \umap/
\index{metric unconditional approximation property!for homogeneous Banach spaces} 
since $\SCE$ does.\vskipb
%n \umbs/ in $\SLP{}$ 
%for all $1\le p<\infty$ as soon as it is an \umbs/ in $\SC{}$.
%Recall further that, by \cite[Th.\ 7]{li96}, 
%$\SLPE$ has complex \umap/ as soon as $\SCE$ 
%has \lmap{1} (and hence complex \umap/).

\dem
Suppose $|n_{j+1}/n_j|\ge q$ for $j\ge l$ and some $q>1$ to 
be fixed later. Let 
$f=\sum a_j\e_{n_j}\in\PTE/$ and $k\ge l$. We show by 
induction that for all $p\ge k$
\begin{equation}\label{positif:hyp}
\|\pi_pf\|_\infty\ge
\biggl(1-\frac{\pi^2}2\frac{1-q^{2(k-p)}}{{q^2}-1}\biggr)
\|\pi_kf\|_\infty+\sum_{j=k+1}^{p}
\biggl(1-\frac{\pi^2}2\frac{1-q^{2(j-p)}}{{q^2}-1}\biggr)
|a_j|.
\end{equation}

\bloc 
There is nothing to show for $p=k$.

\bloc 
%Suppose $(\ref{positif:hyp})$ holds. Let $u\in\T$ 
%be such that
%$\|\pi_pf\|_\infty=|\pi_pf(u)|$. Then there is a $v\in\T$ 
%such that
%\begin{equation}\label{positif:arg}
%\left\{\!\!
%\begin{array}{l}
%|\arg u/v|\le\pi/|n_{p+1}|\\
%|\pi_pf(u)+a_{p+1}\e_{n_{p+1}}(v)|
%=\|\pi_pf\|_\infty+|a_{p+1}|.
%\end{array}\right.
%\end{equation}
%Indeed, there are $|n_{p+1}|$ equidistant 
%points $v\in\T$ such that 
%$a_{p+1}\e_{n_{p+1}}(v)$ and $\pi_pf(u)$ 
%have same argument: there is necessarily one such point 
%at distance at most 
%$\frac 122\pi/|n_{p+1}|=\pi/|n_{p+1}|$.
By 
%\cite[\S I.4, Lemma 8]{me72}, by 
Bernstein's inequality 
applied to $\pi_kf''$ and separately to each 
$a_j\e_{n_j}''$, $j>k$,
\begin{equation}
  \label{positif:bern}
\|\pi_pf''\|_\infty\le n_k^2\|\pi_kf\|_\infty+\sum_{j=k+1}^pn_j^2|a_j|.
\end{equation}
Furthermore, by Lemmas $1$ and $2$ of \cite[\S VIII.4.2]{me72},
\begin{equation}\label{positif:mey}
\|\pi_{p+1}f\|_\infty\ge
\|\pi_pf\|_\infty+|a_{p+1}|-\pi^2/(2n_{p+1}^2)\|\pi_pf''\|_\infty.
\end{equation}
\Ref{positif:mey} together with \Ref{positif:hyp} and \Ref{positif:bern} yield
\Ref{positif:hyp} with $p$ 
replaced by $p+1$. % 
%\begin{eqnarray}
%%
%\nonumber
%|\pi_pf(u)-\pi_pf(v)|&\le&
%\mbox{$\frac12$}|\mbox{$\arg u/v$}|^2\|\pi_pf''\|_\infty\\
%\nonumber
%&\le&\frac{{\pi^2}}{2|n_{p+1}|^2}|n_k|^2
%\|\pi_kf\|_\infty+\sum_{j=k+1}^p
%\frac{{\pi^2}}{2|n_{p+1}|^2}|n_j|^2|a_j|\\
%&\le&\frac{\pi^2}2 q^{2(k-p-1)}\|\pi_kf\|_\infty+\sum_{j=k+1}^p
%\frac{\pi^2}2 q^{2(j-p-1)}|a_j|.\label{positif:BS}
%\end{eqnarray}
%Thus we get successively
%%
%\begin{eqnarray*}
%%
%\lefteqn{\|\pi_{p+1}f\|_\infty\ge|\pi_{p+1}f(v)|}&&\\
%&\ge&|\pi_pf(u)+a_{p+1}\e_{n_{p+1}}(v)|
%-|\pi_pf(u)-\pi_pf(v)|\\
%&\hoch\ge^{(\ref{positif:arg}\&\ref{positif:BS})}&
%\|\pi_pf\|_\infty+|a_{p+1}|-
%\frac{\pi^2}2q^{2(k-p-1)}\|\pi_kf\|_\infty-
%\sum_{j=k+1}^p\frac{\pi^2}2q^{2(j-p-1)}|a_j|\\
%&\hoch\ge^{(\ref{positif:hyp})}&
%\Bigl(1-\frac{\pi^2}2\frac{1-q^{2(k-p-1)}}{q^2-1}\Bigr)
%\|\pi_kf\|_\infty+\sum_{j=k+1}^{p+1}
%\Bigl(1-\frac{\pi^2}2\frac{1-q^{2(j-p-1)}}{q^2-1}\Bigr)
%|a_j|.
%%
%\end{eqnarray*}
%
%\bloc 
%
%So $(\ref{positif:hyp})$ is true for all $p\ge k$. 
%Finally
%
Therefore 
\begin{equation}\label{positif:sidon}
%
\|f\|_\infty=\lim_{p\to\infty}\|\pi_pf\|_\infty\ge
\biggl(1-\frac{\pi^2}2\frac1{{q^2}-1}\biggr)
\biggl(\|\pi_kf\|_\infty+\sum_{j=k+1}^\infty|a_j|\biggr).
%
\end{equation}
%
Thus $\{\pi_j\}_{j\ge k}$ realizes 
\lap{1} with constant 
$1+\pi^2/(2q^2-2-\pi^2)$. As $q$ may be chosen 
arbitrarily large, $E$ has \lmap{1} 
with $\{\pi_j\}$. 
Additionally 
$(\ref{positif:sidon})$ shows by choosing $\pi_kf=0$
that $E$ is a \umbs/ in $\SC{}$.

Finally, the converse holds by Proposition \ref{arith:csq}$(vi)$: 
if $n_{k+1}/n_k$ does not tend to 
infinity while being integer, then there are 
$h\in\Z\setminus\{0,1\}$ and 
an infinite $F$ such that $F,hF\se E$.\eck\vskipb

\remsub 
The technique of Riesz products\index{Riesz product} 
as exposed in \cite[Appendix V,
\S1.II]{ks63} would have sufficed to prove Theorem
\ref{positif:thm}.\vskipb

\remsub 
%
Suppose still that \DEE/ with 
$n_{k+1}/n_k\to\infty$. A 
variation of the above argument yields 
that the space of {\it real\/} 
functions 
with spectrum in $E\cup-E$ has \lap{1}. \vskipb

\remsub 
Note however that there are sets $E$ that satisfy 
$n_{k+1}/n_k\to1$ and nevertheless enjoy 
\I{\infty} (see end of Section \ref{sect:comb}): 
they might be \umbs/ in $\SC{}$, but this is unknown.









\section{An excursion: estimation of the Sidon constant}


The proof of Theorem \ref{positif:thm} furnishes also an estimation of 
the Sidon constant of Hadamard sets. In order to show that this estimate 
is optimal, we undertake first 
the exact computation of the Sidon constant of 
sets with three elements. 


\subsection{Sidon constant of sets with three elements}\label{ss:sidon-constant}

We can compute explicitly the Sidon constant of $\{n_1,n_2,n_3\}$. It 
is equal to 
\begin{equation}
  \label{ubc:ubc}
\max_{r,s>0,\vartheta\in\sT}
(1+r+s)/\|1+r\e_k+\vartheta s\e_l\|_\infty\quad\hbox{with }
\left\{
\begin{array}{l}
k=n_2-n_1\\
l=n_3-n_1.
\end{array}\right.  
\end{equation}
This follows from translation invariance: let
$f=a_1\e_{n_1}+a_2\e_{n_2}+a_3\e_{n_3}$; write $a_i=r_it_i$ with
$r_i\ge0$ and $t_i\in\T$. Then $\|f\|_\infty$ is equal to 
\begin{equation}
\label{ubc:angle}
\begin{array}{c}
r_1\,\|1+r_1^{-1}r_2\e_{n_2-n_1}+\vartheta
%t_1^{(n_3-n_2)/(n_2-n_1)}t_2^{(n_1-n_3)/(n_2-n_1)}t_3
r_1^{-1}r_3\e_{n_3-n_1}\|_\infty\hbox{ where }\vartheta=t'_1t'_2t_3
\hbox{ and}\vspace{3pt}\\
\cases{
t'_1&is any $(n_3-n_2)/(n_2-n_1)$-th power of $t_1$\cr
t'_2&is any $(n_1-n_3)/(n_2-n_1)$-th power of $t_2$.\cr}%\hbox{ with }\vartheta=t'_1t'_2t3
%\&
%\cases{t'_1&any ${n_3-n_2\over n_2-n_1}$-th power of $t_1$\cr
%t'_2&any ${n_1-n_3\over n_2-n_1}$-th power of $t_2$\cr}
\end{array}
\end{equation}
%
%$\vartheta=t'_1t'_2t_3$ and $t'_1$ (\vs $t'_2$) is any
%$(n_3-n_2)/(n_2-n_1)$-th (\vs 
%$(n_1-n_3)/(n_2-n_1)$-th) power of $t_1$ (\vs $t_2$). 
Let us first establish
%\begin{prp}
%%
%Let $k,l\in\Z^*$ be distinct coprime integers. If $|k|,|l|>n$,
%then the complex unconditionality constant of 
%$\{0,k,l\}$ is at most $1+1/n$.
%%
%\end{prp}
%%
%\dem
%This follows from the technique of Riesz products in 
%\cite[Appendix V, \S1.II]{ks63}. Let $K_n$ be the $n$th Fej\'er
%kernel: in particular $K_n\ge0$, $\widehat{K_n}(0)=1$, $\widehat{K_n}(-1)=n/(n+1)$ and
%%$\widehat{K_n}(j)=0$ for $j>n$. Let
%$r,s\ge0$ and $\vartheta\in\T$. Let
%$R=K_n(\e_{k})K_n(\vartheta\e_{l})$. Then $R\ge0$, 
%\begin{eqnarray*}
%\widehat{R}(0)&=& 
%\sum_{\scriptstyle ki+lj=0\atop\scriptstyle |i|,|j|\le n}
%\widehat{K_n}(i)\vartheta ^j\widehat{K_n}(j)=\widehat{K_n}(0)\widehat{K_n}(0)=1,\\
%\widehat{R}(-k)&=&
%\sum_{\scriptstyle ki+lj=-k\atop\scriptstyle |i|,|j|\le n}
%\widehat{K_n}(i)\vartheta ^j\widehat{K_n}(j)=
%\widehat{K_n}(-1)\widehat{K_n}(0)={n\over n+1},\\
%\widehat{R}(-l)&=& 
%\sum_{\scriptstyle ki+lj=-l\atop\scriptstyle |i|,|j|\le n}
%\widehat{K_n}(i)\vartheta ^j\widehat{K_n}(j) =
%\widehat{K_n}(0)\vartheta ^{-1}\widehat{K_n}(-1)={n\over n+1}\vartheta ^{-1}.   
%\end{eqnarray*}
%Therefore $P=1+r\e_{k}+\vartheta s\e_{l}$ satisfies
%$$
%\|P\|_\infty=\sup|P|\int R\,dm\ge\int PR\,dm=\widehat{PR}(0)\ge{n\over n+1}(1+r+s)\eqno{\ecks}$$
%
\begin{lemsub}\label{ubc:lem}
%
Let $(a_{ij})\in\R^2\times\R^3$ and 
$\alpha,\beta\in\R$. Consider
\begin{equation}
  \label{ubs:S}
\left\{\begin{array}{l}
a_{11}\sin\alpha+a_{12}\sin\beta+a_{13}\sin(\beta-\alpha)=0\\
a_{21}\sin\alpha+a_{22}\sin\beta+a_{23}\sin(\beta-\alpha)=0.
\end{array}\right.
\end{equation}
Let $d_1=a_{11}a_{22}-a_{12}a_{21}$, $d_2=a_{11}a_{23}-a_{13}a_{21}$,
$d_3=a_{12}a_{23}-a_{13}a_{22}$. If $d_1d_2d_3\ne0$, then the
solutions to $(S)$ are $\alpha\equiv\beta\equiv0$ mod.\ $\pi$ and, if
furthermore $|d_1|\le|d_2|+|d_3|$, $|d_2|\le|d_1|+|d_3|$ and 
$|d_3|\le|d_1|+|d_2|$,
\begin{equation}\label{ubc:sol}
\cos\alpha={d_3^2-d_1^2-d_2^2\over2d_1d_2}\ ,\
\cos\beta={d_2^2-d_1^2-d_3^2\over2d_1d_3}\ ,\
\cos(\beta-\alpha)={d_1^2-d_2^2-d_3^2\over2d_2d_3},
\end{equation}
where the signs of $\alpha$ and $\beta$ satisfy
$d_2\sin\alpha+d_3\sin\beta=0$. 
%
\end{lemsub}
%
\dem
Suppose that $\alpha\not\equiv0$ or $\beta\not\equiv0$ mod.\ $\pi$. As
\Ref{ubs:S} is equivalent to  
\begin{equation}
  \label{ubc:lem:eq}
(a_{i1}-a_{i3}\cos\beta)\sin\alpha+(a_{i2}+a_{i3}\cos\alpha)\sin\beta=0\
:\ i=1,2,  
\end{equation}
%But then either $\alpha\equiv\beta\equiv0$ mod.\ $\pi$ or
$$(a_{11}-a_{13}\cos\beta)(a_{22}+a_{23}\cos\alpha) =
(a_{21}-a_{13}\cos\beta)(a_{12}+a_{13}\cos\alpha).$$
This simplifies to 
\begin{equation}
  \label{ubc:lem:eq2}
d_1+d_2\cos\alpha+d_3\cos\beta=0
\end{equation}
and by \Ref{ubc:lem:eq},
$$%  \label{ubc:eq2}
\left\{
  \begin{array}{l}
(d_2\sin\alpha+d_3\sin\beta)(a_{i2}+a_{i3}\cos\alpha) = 0\\
(d_2\sin\alpha+d_3\sin\beta)(a_{i1}-a_{i3}\cos\alpha) = 0
  \end{array}\right.:\ i=1,2. 
$$
Therefore 
\begin{equation}
  \label{ubc:lem:eq3}
d_2\sin\alpha+d_3\sin\beta=0:
\end{equation}
otherwise
%$a_{i2}+a_{i3}cos\alpha=a_{i1}-a_{i3}cos\alpha=0$ 
$d_1=d_2=d_3=0$. Equations \Ref{ubc:lem:eq2} and \Ref{ubc:lem:eq3}
yield solution \Ref{ubc:sol}.\eck


\begin{lemsub}\label{ubc:prp}
%
Let $r,s>0$, $k,l\in\Z^*$ distinct and coprime. Let
\begin{eqnarray*}
\Phi(t,\vartheta)&=&|1+r\e^{\ii kt}+\e^{\ii\vartheta}s\e^{\ii lt}|^2\\
&=&1+r^2+s^2+2r\cos(kt)+2s\cos(lt+\vartheta)+2rs\cos((l-k)t+\vartheta).
\end{eqnarray*}
Let $\Phi^*(\vartheta)=\max\limits_{0\le t<2\pi}\Phi(t,\vartheta)$. Then
$\min\limits_{0\le\vartheta<2\pi}\Phi^*(\vartheta)=\Phi^*(\pi/k)$. 
%
\end{lemsub}
%
\dem
Let us first locate the extremal points of
$\Phi$. $\nabla\Phi(t,\vartheta)=0$ is Equation 
\Ref{ubs:S} with 
$$(a_{ij})=\pmatrix{
rk&sl&rs(l-k)\cr
0&s&rs\cr}\quad\hbox{and}\quad\cases{\alpha=kt&\cr\beta=lt+\vartheta.\cr}
$$ 
In the notation of Lemma \ref{ubc:lem}, $d_1=rsk\ne0$, $d_2=r^2sk\ne0$,
$d_3=rs^2k\ne0$. The solution \Ref{ubc:sol}, if it exists, yields
$\Phi(t,\vartheta)=0$ and corresponds to the absolute minimum of
$\Phi$. Every other extremal point of $\Phi$ satisfies 
$$
  \label{ubc:lem:cons}
kt\equiv lt+\vartheta\equiv0\hbox{ mod.\ }\pi. 
$$

$\Phi^*$ is continuous (see \cite[Chapter 5.4]{ps72}) and
$(2\pi/k)$-periodical: choose $j\in\Z$ such that $jl\equiv1$ mod.\
$k$. Then
$$\Phi(t+2j\pi/k,\vartheta) =
|1+r\e^{\ii kt}+s\e^{\ii(\vartheta+2\pi jl/k)}\e^{\ii lt}|^2 = 
\Phi(t,\vartheta+2\pi/k).$$ 
%and by B\'ezout's Theorem, 
%$\Phi(t+2j\pi/k,\vartheta)=\Phi(t,\vartheta+2\pi/k)$ for some
%$j\in\Z$. 
Furthermore 
$\Phi(t,-\vartheta)=\Phi(-t,\vartheta)$ and $\Phi^*$ is even. Thus
$\Phi^*$ attains its minimum on $[0,\pi/k]$.

Let us show that $\Phi^*$ has a local minimum in $\pi/k$ for all
values of $r,s>0$ except eventually one value of $s$ for a given
$r$. Let $t^*$ be such that $\Phi(t^*,\pi/k)=\Phi^*(\pi/k)$. As
before, for $j$ such that $jl\equiv1$ mod.\ $k$,
$$\Phi(t^*,\pi/k+\vartheta) = \Phi(-t^*,-\pi/k-\vartheta) =
\Phi(-t^*+2j\pi/k,\pi/k-\vartheta).$$ 
If $\partial\Phi/\partial\vartheta(t^*,\pi/k)\ne0$, this shows that
$\Phi^*$ has a local minimum in $\pi/k$. Let us suppose that
$\partial\Phi/\partial\vartheta(t^*,\pi/k)=0$. Then 
$\nabla\Phi(t^*,\pi/k)=0$ and 
%We also have
%$\partial\Phi/\partial t(t^*,\pi/k)=0$. This amounts to
%$$
%\left\{\begin{array}{@{}r@{}r@{}r@{}l}
%%\hfil#&\hfil#&\hfil#\cr&#\hfil\cr
%rk\sin kt^*+s&l\sin(lt^*+\pi/k)+rs&(l-k)\sin((l-k)t^*+\pi/k)=0&\\
%            s& \sin(lt^*+\pi/k)+rs&     \sin((l-k)t^*+\pi/k)=0&.
%\end{array}
%\right.
%$$
%In the notation of Lemma \ref{ubc:lem}, $d_1=rsk\ne0$, $d_2=r^2sk\ne0$,
%$d_3=rs^2k\ne0$. As the solution \Ref{ubc:sol} yields
%$\Phi(t^*,\pi/k)=0$ and corresponds to a minimum, 
therefore $kt^*=j\pi$ and $lt^*+\pi/k=j'\pi$ for some
$j,j'\in\Z$. Then $j$ or $j'$ must be odd. 
We have 
\begin{equation}\label{ubc:d2phi}
{\partial^2\Phi\over\partial t^2}(t^*,\pi/k) =
-2\bigl(rk^2(-1)^j+sl^2(-1)^{j'}+rs(l-k)^2(-1)^{j+j'}\bigr).
\end{equation}
We shall suppose that $\partial^2\Phi/\partial t^2(t^*,\pi/k)\ne0$,
which removes at most one value of $s$ for a given $r$. Then, by
the Theorem of Implicit Functions, there is a unique differentiable
function $t_*$ defined  in a neighbourhood of $\pi/k$ such
that 
$$t_*(\pi/k)=t^*\ ,\ \partial\Phi/\partial
t(t_*(\vartheta),\vartheta)=0\ ,\ \partial^2\Phi/\partial
t^2(t_*(\vartheta),\vartheta)<0.$$  
Let
$\Phi_*(\vartheta)=\Phi(t_*(\vartheta),\vartheta)$. Then we have
$\Phi_*'(\pi/k)=\partial\Phi/\partial\vartheta(t^*,\pi/k)=0$ and a
computation yields 
$$\Phi_*''(\pi/k) = { 
{\partial^2\Phi\over\partial t^2} {\partial^2\Phi\over\partial\vartheta^2} -
{\partial^2\Phi\over\partial t\partial\vartheta}^2 \over 
{\partial^2\Phi\over\partial t^2}} (t^*,\pi/k) = 
4rsk^2{{\partial^2\Phi\over\partial t^2}(t^*,\pi/k)}^{-1}\Delta,$$
where $\Delta=((-1)^{j+j'}+r(-1)^{j'}+s(-1)^j)$.
Let us prove that $\Delta<0$ and thus
$\Phi_*''(\pi/k)>0$ and that therefore $\Phi_*$ and consequently
$\Phi^*$ have a local minimum in
$\pi/k$. If we had $\Delta\ge0$ and

\bloc $j$ even, $j'$ odd: then $-1-r+s\ge0$ and by \Ref{ubc:d2phi}
$${\partial^2\Phi\over\partial t^2}(t^*,\pi/k) \ge
2\bigl(-rk^2+(1+r)l^2+r(1+r)(l-k)^2\bigr)=2(r(l-k)+l)^2\ge0;$$ 

\bloc $j$ odd, $j'$ even: then $-1+r-s\ge0$ and by \Ref{ubc:d2phi}
$${\partial^2\Phi\over\partial t^2}(t^*,\pi/k) \ge
2\bigl((1+s)k^2-sl^2+s(1+s)(l-k)^2\bigr)=2(s(l-k)-k)^2\ge0;$$  

\bloc $j$ odd, $j'$ odd: then $1-r-s\ge0$. Considering
\Ref{ubc:d2phi}, we have the following alternative. 
If $l^2\ge r(l-k)^2$, then 
$\partial^2\Phi/\partial t^2(t^*,\pi/k) \ge0$; otherwise
$${\partial^2\Phi\over\partial t^2}(t^*,\pi/k) \ge 
2\bigl(rk^2+(1-r)(l^2-r(l-k)^2)\bigr)=2(r(l-k)-l)^2\ge0.$$
Let us show that then $\Phi^*$ must decrease on $[0,\pi/k]$. Otherwise
there are $0\le\vartheta_1<\vartheta_2\le\pi/k$ such that
$\Phi^*(\vartheta_2)>\Phi^*(\vartheta_1)$. As $\pi/k$ is a local
minimum, there is a $\vartheta_1<\vartheta^*<\pi/k$ such that 
$$\Phi^*(\vartheta^*) =
\max_{\vartheta_1\le\vartheta\le\pi/k}\Phi^*(\vartheta) =
\max_{\scriptstyle          0\le t        < 2\pi\atop
      \scriptstyle\vartheta_1\le\vartheta\le \pi/k}
\Phi(t,\vartheta),$$
\ie there further is some $t^*$ such that $\Phi$ has a local maximum
in $(t^*,\vartheta^*)$. But then 
% and $\nabla\Phi(t^*,\vartheta^*)=0$. By Lemma
%\ref{ubc:lem}, this implies that 
$kt^*\equiv lt^*+\vartheta^*\equiv0$ mod.\ $\pi$
% because the solution \Ref{ubc:sol} yields
%$\Phi(t^*,\vartheta^*)=0$ and corresponds to the absolute minimum of
%$\Phi$. But then 
and $\vartheta^*\equiv0$ mod.\ $\pi/k$ and this is
impossible. That shows the proposition, except for one
value of $s$ at most for a given $r$. But $\Phi^*$ is a continuous
function of $s$ and the proposition is true by a
perturbation.\eck\vskipb

%By Proposition \ref{ubc:prp}, \Ref{ubc:angle} and \Ref{ubc:ubc}, we have
%\begin{thm}
%%
%Let $n_1,n_2,n_3\in\Z$ be distinct. Then the Sidon constant of
%$\{{n_1},{n_2},{n_3}\}$ in $\SC{}$ is
%\begin{equation}\label{ubc:value}\max_{r,s>0}
%{1+r+s\over\|\e_{n_1}+\delta r\e_{n_2}+\epsilon s\e_{n_3}\|_\infty}
%\end{equation}
%with $\delta$ and $\epsilon$ real signs such that
%
%\bloc $\delta=-1$ and $\epsilon=\pm1$ if $2^j\mid n_2-n_1$ and
%$2^j\nmid n_3-n_1$ for some $j$;
%
%\bloc $\delta=\pm1$ and $\epsilon=-1$ if $2^j\mid n_3-n_1$ and
%$2^j\nmid n_2-n_1$ for some $j$;
%
%\bloc $\delta=-1$ and $\epsilon=1$, or $\delta=1$ and $\epsilon=-1$, in
%the remaining case.
%
%In particular, the real and complex unconditionality constants of
%$\{{n_1},{n_2},{n_3}\}$ in $\SC{}$ coincide.
%%
%\end{thm}
%%\dem We may suppose $n_1<n_2<n_3$. Note that in each case 
%%$$\|\e_{n_1}+(-1)^jr\e_{n_2}+(-1)^{j'}s\e_{n_3}\|_\infty=
%%\|1+r\e_{n_2-n_1}+\e^{\ii\pi/(n_2-n_1)}\e_{n_3-n_1}\|_\infty.
%%\eqno{\ecks}$$
%
\exasub
The real and complex unconditionality constant of
$\{0,1,2\}$ in $\SC{}$  is $\sqrt{2}$. Indeed, a case study shows that
$$\|1+r\e_1-s\e_2\|_\infty = 
\cases{r+|s-1|&if $r|s-1|\ge4s$\cr
(1+s)(1+r^2/(4s))^{1/2}&if $r|s-1|\le4s$\cr}$$
and this permits to compute the maximum \Ref{ubc:ubc}, which is obtained
for $r=2$, $s=1$.\vskipb

\exasub
The real and complex unconditionality constant of
$\{0,1,3\}$ in $\SC{}$ is $2/\sqrt{3}$. Indeed, a case study shows that
$\|1+r\e_1-s\e_3\|_\infty$ makes 
$$\cases{
1+r-s&if $s\le r/(4r+9)$\cr
\bigl({2\over27}s(r^2+9+3r/s)^{3/2}
-{2\over27}r^3s+{2\over3}r^2+rs+s^2+1\bigr)^{1/2}
&if $s\ge r/(4r+9)$\cr}$$
and this permits to compute the maximum \Ref{ubc:ubc}, which is obtained
for $r=3/2$, $s=1/2$.\vskipb

These examples are particular cases of the following theorem.

\begin{thmsub}\label{ubc:THM}
Let $n_1,n_2,n_3\in\Z$ distinct such that $n_2-n_1$ and $n_3-n_1$
are coprime. Let $n=\max|n_i-n_j|$. Then the 
Sidon\index{Sidon set!constant} constant of
$E=\{n_1,n_2,n_3\}$ is 
$$\Bigl(\cos{\pi\over2n}\Bigr)^{-1}.$$
%
\end{thmsub}
\dem
We may suppose $n_1<n_3<n_2$. Let $k=n_2-n_1$ and $l=n_3-n_1$. 
By Lemma \ref{ubc:prp},
the Arithmetic-Geometric Mean Inequality bounds the Sidon constant $C$ of
$\{n_1,n_2,n_3\}$ in the following way:
\begin{eqnarray*}
C=\max_{r,s>0}{1+r+s\over\|1+r\e_k+\e^{\ii\pi/k}s\e_l\|_\infty}&\le&
\max_{r,s>0}{1+r+s\over|1+r+\e^{\ii\pi/k}s|}\\&=&
\max_{r,s>0}\biggl(1-\sin^2{\pi\over2k}{4s(1+r)\over(1+r+s)^2}\biggr)^{-1/2}\\
&\le&
\bigl(1-\sin^2(\pi/2k)\bigr)^{-1/2}=\bigl(\cos(\pi/2k)\bigr)^{-1}.
\end{eqnarray*}
This inequality is sharp: we have equality for $r=l/(k-l)$ and
$s=1+r$. In fact the derivative of 
$|1+r\e^{\ii kt}+\e^{\ii\pi/k}s\e^{\ii lt}|^2$ is
then
$$
{8lk\over l-k}\sin{kt\over2}\cos{lt+\pi/k\over
  2}\cos{(l-k)t+\pi/k\over2}
$$
and its extremal points are 
$$
{2j\over k}\pi\ ,\ {2j+1\over l}\pi-{\pi\over lk}\ ,\ {2j+1\over
  l-k}\pi-{\pi\over (l-k)k}\ :\ j\in\Z$$
so that its extremal values are
$$4s^2\cos^2{2j+1\over2k}\pi\ ,\ 4r^2\sin^2{2j+k+1\over2l}\pi\ ,\ 
  4\sin^2{2j+k+1\over2(l-k)}\pi\ :\ j\in\Z.$$
Therefore the maximum of $|1+r\e^{\ii kt}+\e^{\ii\pi/k}s\e^{\ii lt}|^2$ is
$4s^2\cos^2(\pi/(2k))$. \eck\vskipb
 
This proof and \Ref{ubc:angle} yield also the
  more precise 
\begin{prpsub}
The Sidon constant of $\{n_1,n_2,n_3\}$ is attained for 
$$
\vartheta_1\,|n_2-n_3|\,\e_{n_1}+
\vartheta_2\,|n_1-n_3|\,\e_{n_2}+
\vartheta_3\,|n_1-n_2|\,\e_{n_3}
$$
with $\vartheta_1,\vartheta_2,\vartheta_3\in\{-1,1\}$ real signs such
that

\bloc 
$\vartheta_1\vartheta_2=-1$ if $2^j\mid n_2-n_1$ and
$2^j\nmid n_3-n_1$ for some $j$;

\bloc 
$\vartheta_1\vartheta_3=-1$ if $2^j\nmid n_2-n_1$ and
$2^j\mid n_3-n_1$ for some $j$;

\bloc 
$\vartheta_2\vartheta_3=-1$ otherwise.

Thus complex and real unconditionality constants of $E$ in $\SC{}$ coincide.
\end{prpsub}

Let us also underline the following easy consequences of our computation. 
\begin{cor}
\bm{i} 
The Sidon constant of sets with three elements is at most $\sqrt{2}$.

\bm{ii}
The Sidon constant of $\{0,n,2n\}$ is $\sqrt{2}$, while the Sidon constant of 
$\{0,n+1,2n\}$ is at most 
$\bigl(\cos(\pi/(2n)\bigr)^{-1}=1+\pi^2/(8n^2)+o(n^{-2})$.
\end{cor}



\subsection{Sidon constant of Hadamard sets}








Recall that \DEE/ is a Hadamard\index{Hadamard set} set if there is a 
$q>1$ such that $n_{k+1}/n_k\ge q$ for all $k$. It is a 
classical fact that then $E$ is a Sidon set: 
Riesz products\index{Riesz product} (see \cite[Chapter 2]{lr75}) 
even yield effective 
bounds for its Sidon constant. In particular, if $q\ge3$, then $E$'s
Sidon constant is at most $2$. Our computations 
provide an alternative proof for 
$q>\sqrt{\pi^2/2+1}\approx 2.44$ and give a better bound for 
$q>\sqrt{\pi^2+1}\approx 3.30$.
Putting $k=1$ in $(\ref{positif:sidon})$ 
and using Theorem \ref{ubc:THM}, we obtain
%
\begin{corsub}\label{positif:cor}
\index{Sidon set!constant}%
Let \DEE/.

\bm{i} Let $q>\sqrt{\pi^2/2+1}$. If $|n_{k+1}|\ge q|n_k|$ for all $k$, then
the Sidon constant of $E$ is at most $1+\pi^2/(2q^2-2-\pi^2).$

\bm{ii} Let $q\ge2$ be an integer. If $E\supseteq\{n,n+k,n+qk\}$ for some $n$ and $k$, 
then the Sidon constant of $E$ is at least 
$\bigl(\cos(\pi/2q)\bigr)^{-1}\ge1+\pi^2/(8q^2)$.
%
\end{corsub}
In particular, we have the following bounds for the Sidon constant $C$ of $G=\{j^k\}$, $j\in\Z\setminus\{-1,0,1\}$:
$$
1+\pi^2/(8(j+1)^2)\le C\le 
1+\pi^2/(2j^2-2-\pi^2).$$
%
%This estimate is optimal for large $q$ in the following sense: 
%Let $E=\{0,1,q\}$ with $q\ge2$ an integer. 
%Then
%$$
%|1+\e_1(t)-\e_q(t)|^2=3+2\cos t-2\cos(q-1)t-2\cos qt
%$$
%and, as $\cos(q-1)t+\cos qt\ge0$ for 
%$|t|\le\pi/2q$ and $\cos t\le\cos\pi/2q$ 
%for $|t|\in[\pi/2q,\pi]$,
%\begin{eqnarray*}
%\|1+\e_1-\e_q\|_\infty^2
%&\le&5\vee\biggl(3+2\cos\frac\pi{2q}+2+2\biggr)
%=7+2\cos\frac\pi{2q}\\
%&\le&7+2\biggl(1-\frac{16-8\sqrt{2}}{\pi^2}
%\biggl(\frac\pi{2q}\biggr)^2\biggr)
%=9-\frac{8-4\sqrt{2}}{q^2}
%\end{eqnarray*}
%since $\cos t\le1-(16-8\sqrt{2})/{\pi^2}\cdot t^2$
%for $|t|\le\pi/4$. So $E$'s real unconditionality constant is at least
%$$
%3\biggl(9-\frac{8-4\sqrt{2}}{q^2}\biggr)^{-1/2}
%\ge1+\frac{4-2\sqrt2}{9q^2}
%$$
%since
%$(1-t)^{-1/2}\ge1+t/2$ for $t<1$. 
%In fact a bootstrapping argument shows that $E$'s real
%unconditionality constant is at least 
%$1+(\pi^2/12) q^{-2}+o(q^{-2})$, so that Corollary \ref{positif:cor}
%cannot be improved upon this.
%
%We shall now prove that the estimate of Lemma \ref{positif:cor} is of
%the right order by 
%studying the unconditionality constant of sets of three elements in
%$\SC{}$









\section{Density conditions}
\label{sect:comb}










We apply combinatorial tools to find out how ``big'' 
a set $E$ may be while enjoying \I{n} or \J{n}, 
and how ``small'' it must be.

The coarsest notion of largeness is that of 
density. Recall that the maximal density\index{maximal density} of \DE/ 
is defined by 
$$
d^*(E)=\lim_{h\to\infty}\,\max_{\vphantom{h}a\in\sZ}
\frac{\mes{E\cap\{a+1,\dots,a+h\}}}h.
$$

Suppose $E$ enjoys \I{n}\index{maximal density!of independent sets} 
with $n\ge2$. Then $E$ is a 
\EL{2n} set by Theorem \ref{mub:thm}$(i)$. By 
\cite[Th.\ 3.5]{ru60} (see also \cite[\S1, Cor.\ 2]{mi75}), $d^*(E)=0$. 
Now suppose $E$ enjoys complex or real 
\J{n} with $n\ge2$. As Li\index{Li, Daniel} \cite[Th.\ 2]{li96} 
shows, there are sets $E$ 
such that $\SCE$ has \lmap{1} while $E$ contains 
arbitrarily 
long arithmetic sequences: we cannot apply Szemeredi's Theorem.

Kazhdan\index{Kazhdan, David A.} (see \cite[Th.\ 3.1]{hi82}) 
proved that if $d^*(E)>1/n$, 
then there is a $t\in\{1,\dots,n-1\}$ such that 
$d^*(E\cap E+t)>0$. One 
might hope that it should in fact suffice to choose 
$t$ in any interval of length $n$. 
However, Hindman\index{Hindman, Neil} \cite[Th.\ 3.2]{hi82} 
exhibits a counterexample: 
given $s\in\Z$ and positive 
$\eps$, there is a set $E$ with 
$d^*(E)>1/2-\eps$ and 
there are arbitrarily large $a$ such 
that $E\cap E-t=\emptyset$ for all 
$t\in\{a+1,\dots,a+s\}$. Thus, we have to 
be satisfied with
%
\begin{lem}\label{com:lem} 
%
  Let \DE/ with positive maximal density.  Then there is a $t\ge1$
  such that the following holds: for any $s\in\Z$ we have some $a$,
  $|a|\le t$, such that $d^*(E+a\cap E+s)>0$.
%
\end{lem}
% 
\dem 
By a result of Erd\H{os}\index{Erd\H{o}s, Paul} 
(see \cite[Th.\ 3.8]{hi82}), there 
is a $t\ge1$ such that $F=E+1\cup\dots\cup E+t$ 
satisfies $d^*(F)>1/2$. But then, by \cite[Th.\ 3.4]{hi82}, 
$d^*(F\cap F+s)>0$ for any $s\in\Z$. 
This means that for 
any $s$ there are $1\le a,b\le t$ such that 
$d^*(E+a\cap E+s+b)>0$. 
\eck\vskipb

We are now able to prove 
%
\begin{prp}\label{comb:thm}
  \index{maximal density!of block independent sets}
%
  Let \DE/.
  
  \bm{i} If $E$ has positive maximal density, then there is an
  $a\in\Z$ such that $E\cup\{a\}$ fails real \J{2}.  Therefore $E$
  fails real \J{3}.
  
  \bm{ii} If $d^*(E)>1/2$, then $E$ fails real \J{2}.
%
\end{prp}
%
\dem 
$(ii)$ is established in \cite[Prop.\ 14]{li96}. 
$(i)$ is a consequence 
of Lemma \ref{com:lem}: indeed, if 
$E$ has positive maximal density, then this lemma yields 
some 
$a\in\Z$ and an infinite $F\se E$ such that 
for all $s\in F$ there are arbitrarily large 
$k,l\in E$ such that $k+a=l+s$. Thus 
$E\cup\{a\}$ 
fails real \J{2}. Furthermore, $E$ fails 
real \J{3} by Proposition \ref{arith:csq}$(iv)$.
\eck\vskipb

\rem 
We may reformulate the remaining open case of \J{2}. 
Let us introduce the infinite difference\index{infinite difference set}
set of $E$: 
$\Delta E=\{t:\mes{E\cap(E-t)}=\infty\}$ (see 
\cite{st79} and \cite{ru78}). Then $E$ has real \J{2} if and only if, 
for any $a\in E$, $\Delta E$ meets $E-a$ 
finitely many times only. 
Thus our question is: are there sets with 
positive maximal density such that 
$E-a\cap\Delta E$ is finite for all $a\in E$~? \vskipb

Proposition \ref{comb:grow} 
and Theorem \ref{positif:thm} show that there is 
only one general condition of lacunarity on $E$ that 
ensures properties \I{n}, \J{n} or \I{\infty}, \J{\infty}: 
$E$ must grow exponentially\index{exponential growth} or 
superexponentially\index{superexponential growth}.
One may nevertheless 
construct inductively 
``large'' sets that enjoy these properties: 
they must only be 
sufficiently irregular to avoid all arithmetical relations. 
Thus there are sequences with growth slower than 
$k^{2n-1}$\index{polynomial growth} which nevertheless 
enjoy both \I{n} and complex and real 
\J{n}. See \cite[\S II, (3.52)]{hr83} 
for a proof in the case $n=2$: 
it can be easily adapted to 
$n\ge 2$ and shows also the way to construct, for any sequence
$n_k\to\infty$, 
sets that satisfy \I{\infty} and \J{\infty} and grow more 
%
slowly than $k^{n_k}$\index{superpolynomial growth}.









\section{Unconditionality vs.\ probabilistic indepen\-d\-en\-ce}
\label{sect:proba}








\subsection{Cantor group}







Let us first show how simple the 
problems of \umbs/ and \umap/ become when considered 
for independent uniformly distributed random variables 
and their span in some space.

Let $\D^\infty$ be the Cantor group\index{Cantor group} and $\Gamma$ 
its dual group of Walsh functions. 
Consider the set $R=\{r_i\}\se\Gamma$ 
of Rademacher\index{Rademacher functions} functions, \ie 
of the coordinate functions on $\D^\infty$: 
they form a family of independent random variables that 
take values $-1$ and $1$ with equal probability $\frac12$: 
Thus $\|\sum\epsilon_ia_ir_i\|_X$ does not depend on the choice of 
signs $\epsilon_i=\pm1$ for any 
homogeneous Banach space $X$ on $\D^\infty$ 
%space 
%$X\in\{\script{C}(\D^\infty),{\fam0 L}^p(\D^\infty):1\le p<\infty\}$ and 
and $R$ is a real $1$-\ubs/
\index{$1$-unconditional basic sequence of characters!on the Cantor group}
in $X$. 

Clearly, $R$ is 
also a complex \ubs/ in all such $X$. But its complex unconditionality
constant is $\pi/2$\index{real vs.\ complex}\index{complex vs.\ real} 
\cite{se97} and ${\fam0 L}^p_W(\D^\infty)$ has complex \umap/
if and only if $p=2$
\index{metric unconditional approximation property!on the Cantor group}
or $W=\{w_i\}\se\Gamma$ is finite. 
Indeed, $W$ would have an analogue property \UP/ of 
block unconditionality in ${\fam0 L}^p(\D^\infty)$: for any $\eps>0$ 
there would be $n$ such that 
$$
\maxl_{w\in\sT}\|\epsilon a w_1+b w_n\|_p
\le(1+\eps)\|a w_1+b w_n\|_p.
$$
But this is false: 
for $1\le p<2$, take $a=b=1$, $\epsilon=\hbox{i}$:
$$
\maxl_{\epsilon\in\sT}\|\epsilon w_1+ w_n\|_p\ge
\bigl(\hbox{$\frac12$}
(|\hbox{i}+1|^p+|\hbox{i}-1|^p)\bigr)^{1/p}
=\sqrt{2}>\| w_1+ w_n\|_p=2^{1-1/p};
$$
for $2<p\le\infty$, take $a=1$, $b={\fam0 i}$, $\epsilon=\hbox{i}$:
$$
\maxl_{\epsilon\in\sT}\|\epsilon w_1+{\fam0 i} w_n\|_p\ge
\bigl(\hbox{$\frac12$}(|{\fam0 i}+{\fam0 i}|^p+
|{\fam0 i}-{\fam0 i}|^p)\bigr)^{1/p}
=2^{1-1/p}>\| w_1+{\fam0 i} w_n\|_p=\sqrt{2}.
$$

This is simply due to the fact that the image domain 
of the characters on $\D^\infty$ is 
too small. Take now 
the infinite torus $\T^\infty$ 
and consider the 
set $S=\{s_i\}$ of Steinhaus functions, 
\ie the coordinate 
functions on $\T^\infty$: 
they form again a family of independent 
random variables with values uniformly distributed in $\T$. 
Then $S$ is clearly a complex 
$1$-\ubs/ 
\index{$1$-unconditional basic sequence of characters!on the infinite torus}
in any homogeneous Banach space $X$ on $\T^\infty$.
% $X\in\{\script{C}(\T^\infty),
%{\fam0 L}^p(\T^\infty)\,(1\le p<\infty)\}$.









\subsection{Two notions of approximate probabilistic independence}
\label{ss:proba:two}







As the random variables $\{\e_n\}$ also have their 
values uniformly distributed in $\T$, 
some sort of approximate independence 
should suffice to draw 
the same conclusions as in the case of $S$.

A first possibility is to look at the joint 
distribution of 
$(\e_{p_1},\dots,\e_{p_n})$, 
$\lst pn\in E$, and to 
ask it to be close to the product of the distributions 
of the $\e_{p_i}$. For example, 
Pisier\index{Pisier, Gilles}
\cite[Lemma 2.7]{pi83} 
gives the following characterization:
$E$ is a Sidon\index{Sidon set} set if and only if there are a 
neighbourhood $V$ of $1$ 
in $\T$ and 
$0<\varrho<1$ such that for any finite $F\se E$
\begin{equation}\label{proba:pisier}
m[\e_p\in V:p\in F]\le\varrho^{\smes{F}}.
\end{equation}
Murai \index{Murai, Takafumi}\cite[\S4.2]{mu82} 
calls 
\DE/ pseudo-independent\index{pseudo-independent set} if for all $\lst An\se\T$
\begin{equation}\label{proba:pi}
m[\e_{p_i}\in A_i:1\le i\le n]
\tol_{
\scriptstyle p_i\in E\atop
\scriptstyle p_i\to\infty}
\prod_{i=1}^n
m[\e_{p_i}\in A_i]=\prod_{i=1}^nm[A_i].
\end{equation}
We have
%His theorem \cite[Lemma 30]{mu82} gives together 
%with Proposition \ref{arith:lim}$(iii)$ the following 
%
\begin{prpsub}\label{proba:murai}
%
  Let \DE/.  The following are equivalent. 

\bm{i} 
$E$ is pseudo-independent

\bm{ii}
$E$ enjoys
  \I{\infty}.

\bm{iii}
For every $\eps>0$ and $m\ge1$, there is a finite subset $G\se E$ such
that the Sidon constant of any subset of
$E\setminus G$ with $m$ elements is less than $1+\eps$. 
%
\end{prpsub}
%
Note that by Corollary \ref{arith:cor}, 
$(\ref{proba:pi})$ does not imply $(\ref{proba:pisier})$.

\dem
$(i)\ssi(ii)$  follows by Proposition \ref{arith:lim}$(iii)$ and
\cite[Lemma 30]{mu82}. $(iii)\imp(ii)$ is true because $(iii)$ is just
what is needed to draw our conclusion in Corollary
\ref{arith:cor}. Let us prove $(i)\imp(iii)$. Let $\eps>0$, $m\ge1$
and $\script{A}$ be a covering of $\T$ with intervals of length
$\eps$. By \Ref{proba:pi}, there is a finite set $G\se E$ such that
for $\lst pm\in E\setminus G$ and $A_i\in\script{A}$ we have 
$m[\e_{p_i}\in A_i:A_i\in\script{A}]>0$. But then 
$$\Bigl\|\sum a_i\e_{p_i}\Bigr\|_\infty\ge\sum|a_i|\cdot(1-\eps).\eqno{\ecks}$$

\remsub
$(ii)\imp(iii)$ may be proved directly by the 
technique of Riesz products\index{Riesz product}: 
see \cite[Appendix V, \S1.II]{ks63}.\vskipb

Another possibility is to define some notion of 
almost independence. Berkes \cite{be87} introduces 
the following notion: let us call a sequence 
of random variables $\{X_n\}$ almost 
i.i.d.\index{almost i.i.d. sequence} 
(independent and 
identically distributed) if, after enlarging 
the probability space, there is an i.i.d.\ sequence $\{Y_n\}$ 
such that $\|X_n-Y_n\|_\infty\to0$. We have the straightforward 
%
\begin{prpsub}\label{proba:berkes}
%
  Let \DEE/. If $E$ is almost i.i.d., then $E$ is a Sidon
  set\index{Sidon set!with constant asymptotically $1$} with constant
  asymptotically $1$.
%
\end{prpsub}
%
\dem
%
Let $\{Y_j\}$ be an i.i.d sequence and suppose 
$\|\e_{n_j}-Y_j\|_\infty\le\eps$ for $j\ge k$. 
Then 
$$
\sum_{j\ge k}|a_j|= 
\Bigl\|\sum_{j\ge k}a_jY_j\Bigr\|_\infty\le
\Bigl\|\sum_{j\ge k}a_j\e_{n_j}\Bigr\|_\infty+
\eps\sum_{j\ge k}|a_j|
$$
and the unconditionality constant of 
$\{\lstf nk\}$ is less than $(1-\eps)^{-1}$.
%
\eck\vskipb

Suppose \DEE/ is such that $n_{k+1}/n_k$ 
is an integer for all $k$. 
In that case, Berkes \cite{be87} proves that $E$ is 
almost i.i.d.\ if and only if 
$n_{k+1}/n_k\to\infty$. 
We thus recover a part of Theorem \ref{positif:thm}.

\questsub 
What about the converse in Proposition \ref{proba:berkes}~?












\section{Summary of results. Remarks and questions}
\label{sect:resume}







For the convenience of the reader, we now reorder 
our results by putting together those which are 
relevant to a given class of Banach spaces. 

Let us first summarize our arithmetical results on the geometric sequence
$G=\{j^k\}_{k\ge0}$ ($j\in\Z\setminus\{-1,0,1\}$). The number given in the
first (\vs second, third) column is the value $n\ge1$ for which the set in the
corresponding row achieves exactly \I{n} (\vs complex \J{n}, real
\J{n}).\vskipa



\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|r||c|c|c|} \hline
$G=\{j^k\}_{k\ge0}$ with $|j|\ge2$&\I{n} &\C-\J{n}&\R-\J{n}\\\hline
$G$, $j>0$ odd\vphantom{\LARGE I} &$|j|$ &$|j|$ &$\infty$\\
$G$, $j>0$ even &$|j|$ &$|j|$ &$|j|$ \\
$G\cup\{0\}$, $j>0$ odd &$|j|$ &$|j|-1$ &$\infty$\\
$G\cup\{0\}$, $j>0$ even &$|j|$ &$|j|-1$ &$|j|-1$ \\
$G$, $G\cup\{0\}$, $j<0$ odd &$|j|-1$&$|j|$ &$\infty$\\
$G$, $G\cup\{0\}$, $j<0$ even &$|j|-1$&$|j|$ &$|j|$ \\
$G\cup-G$, $G\cup-G\cup\{0\}$, $j$ odd&$1$&$1$ &$\infty$\\
$G\cup-G$, $j$ even &$1$ &$1$ &$|j|$ \\
$G\cup-G\cup\{0\}$, $j$ even &$1$ &$1$ &$|j|/2$ \\\hline
\end{tabular}\nopagebreak\smallskip\\\nopagebreak
\refstepcounter{thm}Table \thethm
\end{center}\index{geometric sequences}






\subsection[The case $X=\SLP{}$ with $p$ an even integer]
{The case $X={\fam0 L}^p(\hbox{\zwm T})$ with $p$ an even integer} 








Let $p\ge4$ be an even integer. 
We observed the following facts.

\vskipa\bloc 
Real and complex \umap/ differ among subspaces $\SLPE$ for each $p$:
consider Proposition \ref{res:geo} or $\SLPE$ with $E=\{\pm(p/2)^k\}$.

\vskipa\bloc
By Theorem \ref{umap:thm}, $\SLPE$ has complex (\vs real) 
\umap/ if so does $\SLE{p+2}$; 

\vskipa\bloc
The converse is false for any $p$. In the complex case, 
$E=\{(p/2)^k\}$ is a counterexample. In the real 
case, take $E=\{0\}\cup\{\pm p^k\}$.

\vskipa\bloc 
Property \umap/ is 
not stable under unions with an element: for each $p$, 
there is a set $E$ 
such that $\SLPE$ has complex (\vs real) 
\umap/, but $\SLP{E\cup\{0\}}$ does not. 
In the complex case, consider $E=\{(p/2)^k\}$. 
In the real case, consider 
$E=\{\pm(2\lceil p/4\rceil)^k\}$. 

\vskipa\bloc 
If $E$ is a symmetric set and $p\ne2$, then $\SLPE$ fails complex \umap/. 
Proposition \ref{arith:sym} gives a criterion for real \umap/.\vskipb

What is the relationship between \umbs/ and complex \umap/~? We have by 
Proposition \ref{arith:csq}$(i)$ and \ref{umap:prp:fdd}$(i)$
%
\begin{prpsub}\label{resume:mb_um}
%
  Let \DEE/ and $n\ge1$.
  
  \bm{i} If $E$ is a \umbs/ in $\SL{4n-2}{}$, then $\SLE{2n}$ has complex
  \umap/.
  
  \bm{ii} If $\{\pi_k\}$ realizes complex \umap/ in $\SLE{2n}$, then $E$
  is a \umbs/ in $\SL{2n}{}$.
%
\end{prpsub}

We also have, by Proposition \ref{comb:thm}$(i)$
%
\begin{prpsub} 
%
  Let \DE/ and $p\ne2,4$ an even integer. If $\SLPE$ has real \umap/,
  then $d^*(E)=0$.
%
\end{prpsub}

Note also this consequence of Propositions 
\ref{mub:trans}, \ref{arith:trans}, \ref{proba:murai} and Theorems 
\ref{mub:thm}, \ref{umap:thm} 
%
\begin{prpsub} 
%
  Let $\sigma>1$ and $E=\{[\sigma^k]\}$. Then the following properties
  are equivalent:
  
  \bm{i} $\sigma$ is transcendental\index{transcendental numbers};
  
  \bm{ii} $\SLPE$ has complex \umap/ for any even integer $p$;
  
  \bm{iii} $E$ is a \umbs/ in any $\SLP{}$, $p$ an even integer;
  
  \bm{iv} $E$ is pseudo-independent.
  
  \bm{v} For every $\eps>0$ and $m\ge1$, there is an $l$ such that for
  $\lst km\ge l$ the Sidon constant of
  $\{[\sigma^{k_1}],\dots,[\sigma^{k_m}]\}$ is less than $1+\eps$.
%
\end{prpsub}



 
 





\subsection[Cases $X=\SLP{}$ with $p$ not an even integer and $X=\SC{}$]
{Cases $X={\fam0 L}^p(\hbox{\zwm T})$ 
with $p$ not an even integer and $X=\hbox{\zwe C}(\hbox{\zwm T})$} 





In this section, $X$ denotes either 
$\SLP{}$, $p$ not an even integer, or $\SC{}$.

Theorems \ref{mub:thm} and \ref{umap:thm} 
only permit us to use the negative results of Section 
\ref{sect:arith}: thus, we can just gather 
negative results about the functional properties 
of $E$. 
For example, we know by 
Proposition \ref{arith:csq}$(iv)$ that \I{\infty} and \J{\infty} are 
stable under union with an element. Nevertheless, 
we cannot conclude that the same holds 
for \umap/. 
The negative results are (by Section \ref{sect:arith}):

\vskipa\bloc
for any infinite \DE/, 
$X_{E\cup2E}$ fails real \umap/. Thus \umap/ 
is not stable under unions; 

\vskipa\bloc 
if $E$ is a polynomial sequence (see Section \ref{sect:arith}), 
then $E$ is not a 
\umbs/ in $X$ and $X_E$ fails real 
\umap/;

\vskipa\bloc 
if $E$ is a symmetric set, then $E$ is not a 
\umbs/ in $X$ 
and $X_E$ fails complex \umap/. Proposition \ref{arith:sym} 
gives a criterion for real \umap/;

\vskipa\bloc 
if $E=\{[\sigma^k]\}$ with 
$\sigma>1$ an algebraic number ---~in particular 
if $E$ is a 
geometric sequence~---, then $E$ is not a 
\umbs/ in $X$ 
and $X_E$ fails complex \umap/.\vskipb 

Furthermore, by Proposition \ref{res:geo}, 
real and complex \umap/ differ in X. 

Theorem \ref{positif:thm} is the only but general 
positive result on \umbs/ and complex \umap/ in $X$.
Proposition \ref{res:geo} yields further examples 
for real \umap/.

What about the sets that satisfy 
\I{\infty} or \J{\infty}~? We only know that \I{\infty} 
does not even ensure Sidonicity by Corollary \ref{arith:cor}.

One might wonder whether for some reasonable class of sets 
$E$, $E$ is a finite 
union of sets that enjoy \I{\infty} or \J{\infty}. 
This is false 
even for Sidon sets: for example, let $E$ be the geometric
sequence $\{j^k\}_{k\ge0}$ with $j\in\Z\setminus\{-1,0,1\}$ 
and suppose $E=E_1\cup\dots\cup E_n$. 
Then $E_i=\{j^k\}_{k\in A_i}$, where the $A_i$'s are a 
partition of the set of positive integers. But then one 
of the $A_i$ contains arbitrarily large $a$ and $b$ 
such that $|a-b|\le n$. This means that there is an infinite 
subset $B\se A_i$ 
and an $h$, $1\le h\le n$, such that $h+B\se A_i$. 
We may apply Proposition \ref{arith:csq}$(vi)$: 
$E_i$ enjoys neither \I{j^h+1} nor complex 
\J{j^h+1} --- nor real \J{j^h+1} 
if furthermore $j$ is even. 

Does Proposition \ref{resume:mb_um}$(ii)$ remain true for general
$X$~? We do not know this. Suppose however that we know that 
$\{\pi_k\}$ realizes \umap/ in the following strong manner: for any
$\eps>0$, a tail $\{\pi_k\}_{k\ge l}$ is a $(1+\eps)$-unconditional
a.s.\ in $X_E$. 
Then $E$ is trivially a \umbs/ in $X$. In particular, this is the case
if 
$$1+\eps_n=
\sup_{\epsilon\in\sU}\|\Id -(1+\epsilon)\pi_n\|_{\script{L}(X)}$$ 
converges so rapidly to $1$ that $\sum\eps_n<\infty$. Indeed,
$$
\sup_{\epsilon_k\in\sU}
\|\pi_{n-1}+\sum_{k\ge n}\epsilon_k\Delta\pi_k\|
\le(1+\eps_n)
\sup_{\epsilon_k\in\sU}
\|\pi_n+\sum_{k>n}\epsilon_k\Delta\pi_k\|.
$$
and thus, for all $f\in\PTE/$,
$$
\sup_{\epsilon_k\in\sU}
\|\pi_lf+\sum_{k>l}\epsilon_k\Delta\pi_kf\|
\le\prod_{k>l}(1+\eps_k)\,\|f\|.
%\tol_{l\to\infty}\|f\|.
$$

%Is there a result corresponding to Proposition \ref{resume:mb_um}$(ii)$~? We 
%do not know. But suppose that 
%$1+\eps_n=\|\Id -(1+\epsilon)\pi_n\|_{\script{L}(X)}$ 
%converges sufficiently rapidly to $1$: suppose 
%that not only $\eps_n\to0$ but also $\sum\eps_n<\infty$. As 
%$$
%\max_{\epsilon_i\in\sT}
%\biggl\|\sum_{i=l}^k\epsilon_ia_i\e_{n_i}
%+\sum_{i>k}a_i\e_{n_i}\biggr\|_X\le
%(1+\eps_k)\max_{\epsilon_i\in\sT}
%\biggl\|
%\sum_{i=l}^{k-1}\epsilon_ia_i\e_{n_i}
%+\sum_{i\ge k}a_i\e_{n_i}\biggr\|_X,
%$$
%we get 
%$$
%\max_{\epsilon_i\in\sT}\Bigl\|\suml_{i\ge l}\epsilon_ia_i\e_{n_i}\Bigr\|_X
%\le\prodl_{i\ge l}(1+\eps_i)\Bigl\|\suml_{i\ge l}a_i\e_{n_i}\Bigr\|_X,
%$$
%and if $\sum\eps_n<\infty$, then 
%$(1+\eps_l)(1+\eps_{l+1})\cdots\mathop{\tol}\limits_{l\to\infty}1$, \ie 
%$E$ is an \umbs/.

Let us finally state 
%
\begin{prpsub}
%
  Let \DE/. If $X_E$ has real \umap/, then $d^*(E)=0$.
%
\end{prpsub}









\subsection{Questions}








The following questions remain open:

\vskipa{\bf Combinatorics } Regarding Proposition \ref{comb:thm}$(i)$,
is there a set $E$ enjoying \J{2} with positive maximal density, or
even with a uniformly bounded pace~? Furthermore, may a set $E$ with
positive maximal density admit a partition $E=\bigcup E_i$ in finite sets
such that all $E_i+E_j$, $i\le j$, are pairwise disjoint~? Then
$\SLE{4}$ would admit a $1$-unconditional 
\fdd/\index{$1$-unconditional fdd@$1$-unconditional \fdd/!for $\SLE{4}$} 
by Proposition
\ref{umap:prp:fdd}$(i)$.

\vskipa{\bf Functional analysis } Let $X\in\{\SL{1}{},\SC{}\}$ and
consider Theorem \ref{sbd:thm}. Is \UP/ sufficient for $X_E$ to share
\umap/~? Is there a set \DE/ such that some space $\SLPE$, $p$ not an
even integer, has \umap/, while $\SCE$ fails it~?

It could be worthwile to look at certain subsets of $E=\{\sum_{i\in
 F}n_i:F\subset\N\hbox{ finite}\}$ with a very rapidly growing
sequence $\{n_i\}$. By \cite[\S VIII]{me72}, it suffices to study
$E_\infty=\{\prod_{i\in F}s_i:F\subset\N\hbox{ finite}\}$ in the
dual group of $\T^\infty$. Then ${\fam0 L}^1_{E_\infty}(\T^\infty)$ has \uap/
by an argument of D\'echamps-Gondim. Does it have \umap/ ? Does ${\cal
 C}_{E_\infty}(\T^\infty)$ enjoy the Daugavet\index{Daugavet property}
property and thus fail
\uap/ ? Does the natural projection of ${\cal
 C}_{E_\infty}(\T^\infty)$ onto $\script{C}(\D^\infty)$ have a closed
image~?

\vskipa{\bf Harmonic analysis } Is there a Sidon set \DEE/ of constant
asymptotically $1$ such that $n_{k+1}/n_k$ is uniformly bounded~? 
%Is it possible that for any $\eps>0$ there are $n_3\le qn_2\le q^2n_1$
%such that
%$$
%\forall a_1,a_2,a_3\quad
%(1+\eps)\|a_1\e_{n_1}+a_2\e_{n_2}+a_3\e_{n_3}\|_\infty
%\ge|a_1|+|a_2|+|a_3|\ ?$$
What about the case $E=[\sigma^k]$ for a
transcendental\index{transcendental numbers} 
$\sigma>1$~? If $E$ enjoys \I{\infty}, is $E$ a \umbs/
in $\SLP{}$ $(1\le p<\infty)$~? What about \J{\infty}~?
%
%
%
%
%
%\baselineskip=11pt\parsep=0pt \bibliography{references}
%
%
%
%
%
%\end{eqnarray}

\baselineskip=14.6pt minus 2pt
\chapter{Unconditional entry basic sequences}\label{chap2}



\section{Introduction}

We study the following isometric question on the Schatten class
$S^p$. How many matrix coefficients of an operator $x\in S^p$ must
vanish so that the norm of $x$ does not depend on the argument, or on
the sign, of the remaining matrix coefficients~? This is the case if
the remaining nonzero matrix entries are a complex, or real,
$1$-unconditional basic sequence in $S^p$. Thus we are looking for the
isometric counterpart of $\sigma(p)$ sets of entries $I\se\N\times\N$,
which have been introduced recently by Harcharras.

We show that for our purpose, sets of matrix entries $I\se\N\times\N$
are best understood as vertices for polygonal lines that follow
alternatively the horizontal or the vertical axis in $\N\times\N$. If
$p$ is an even integer, we obtain in fact a complete description of
$1$-unconditional basic sequences in $S^p$ by these means (see Th.\ 
\ref{sigma:THM}). The main step is a characterization of rectangular
polygons that return on their track (Th.\ \ref{gon:thm}). Note that as
in the case of sequences of characters in $L^p(\T)$, $p$ an even integer, complex and real
$1$-unconditionality coincide.

We close this chapter with two remarks on Harcharras' 
recent paper \cite{ha98}. In
the first place, we show that c.b.\ $\Lambda(p)$ sets cannot contain
the product of two infinite sets of characters. In the second place,
we point out two new possibilities to construct $\Lambda(p)$ sets.

\vskipa {\bf Notation } $\T=\{t\in\C:\,|t| = 1\}$ is the unit circle
endowed with its Haar measure $dm$ and \Z\ its dual group of integers:
for each $n\in\Z$, let $\e_n(t)=t^n$. The cardinal of \DE/ is written
\mes{E}.

For a space $X$ of integrable functions on \T\ and \DE/, $X_E$ denotes the
space of functions with Fourier spectrum in $E$: $X_E=\bigl\{f\in X:
\widehat{f}(n)=\int\e_{-n}f\,dm=0 \mbox{ if }n\notin E\bigr\}$.

%We shall also denote by $\f_n$, $n\ge0$, the
The Schatten class $S^p$, $1\le p<\infty$, is the space of those
compact operators $x$ on $\ell_2$ such that
$\|x\|_p=(\trace|x|^p)^{1/p}<\infty$. The entry\index{entry} $\e_{rc}$ is the
operator on $\ell_2$ that maps the $c$th basis vector on the $r$th
basis vector. We shall also consider $\e_{rc}$ as 
the indicator function of $(r,c)$ from $\N\times\N$ to $\N$. 
The matrix coefficient $(r,c)$  of an operator $x$ is
$x_{rc}=\trace x\e_{rc}^*$ and its matrix representation is $x\sim\sum
x_{rc}\e_{rc}$.
%, We shall use the matrix
%representation for operators: $x=(x_{rc})_{r,c\ge0}$.
%for $(r,c)\in\N\times\N$, $x_{rc}$ is the entry
%$(r,c)$ of the matrix associated to $x$. 
%We call entries the operators
%$\e_{rc}=(\delta_{rr'}\delta_{cc'})_{r',c'\ge0}$. 
%is the operator
%whose entries all vanish but $(r,c)$, that is $1$. 
A Schur multiplier\index{Schur multiplier}
$T$ on $S^p$ associated to $(\mu_{rc})_{r,c\ge0}$ is a bounded
operator on $S^p$ such that 
$T\e_{rc}=\mu_{rc}\e_{rc}$. 
%that multiplies each entry $(r,c)$ of $x\in S^p$ by
%$\mu_{rc}$. 
$T$ is furthermore completely 
bounded\index{completely bounded Schur multiplier} 
(c.b.\ for short) if
$T\otimes\Id_{S^p}$ is bounded on $S^p(\ell_2\otimes_2\ell_2)\equiv
S^p(S^p)$. 

For \DI/, the entry space\index{entry subspace of $S^p$} 
$S^p_I$ is the space of
those $x\in S^p$ whose matrix representation is supported by $I$:
$x_{rc}=0$ if $(r,c)\notin I$. 
%entries all vanish off $I$. 
$S^p_I$ is also the
closed subspace of $S^p$ spanned by $\{\e_q\}_{q\in I}$.


%
%
%
%
%\section{$1$-unconditional entry subspaces in Schatten classes}
%
%
%
%
%
\section{Rectangular polygons in the two-dimensional integer lattice}
%\hbox{I\kern-3pt N$\times$I\kern-3pt N}
%  $\Bbb{N}\times\Bbb{N}$
%
%
%
%
%
%
This section is of purely combinatorial nature. We introduce and study
the two objects that we need in order to describe $1$-unconditional
basic sequences of matrix entries. 
\begin{dfn}
%
  Let $p=2s\ge0$ be an even integer and \DI/.
  
  \bm{i} A rectangular line\index{rectangular line} $L$ in $I$ is a sequence of vertices
  $(\lst qs)\in I$ such that two successive vertices have a row or a
  column in common. Two lines are equivalent if they are equivalent as
  closed curves.
  
%  \bm{ii} $L$ does not loop if its vertices $q_i$ are all distinct.
%  $L$ is simple if it is, among all equivalent arcs, the rectangular
%  line with the least number of vertices.
  
  \bm{iii} Two rectangular lines $L,L'$ are disconnected if every
  vertex of $L$ has neither row nor column in common with any vertex
  of $L'$. Otherwise they are 
connected\index{connected rectangular lines}.
  
  \bm{iv} A $p$-gon\index{p-gon@$p$-gon} $P$ in $I$ is a rectangular line of $p$ vertices
  in $I$ of the form
  $$
  ((r_1,c_1),(r_1,c_2),(r_2,c_2),(r_2,c_3),\dots,(r_s,c_s),(r_s,c_1)).
  $$
  It is also described by the sequence of $p$ segments of the form
  $$
  (r_1,c_1)\to(r_1,c_2)\downto(r_2,c_2)\to(r_2,c_3)\downto
  \dots\downto(r_s,c_s)\to(r_s,c_1)\downto,
  $$
  where the last arrow $\downto$ indicates the segment
  $(r_s,c_1)\downto(r_1,c_1)$, so that a polygon is a closed curve.
  Thus $P$ is a rectangular line whose odd sides lie in a row and
  whose even sides lie in a column.
%Two polygons are equivalent if they are a permutation of each other.
  
  \bm{v} Let $\Alpha_s^I=\{\alpha\in\N^I:\sum_{q\in I}\alpha_q=s\}$.
  The set of $p$-gonal relations\index{p-gonal relation@$p$-gonal relation} in $I$ is
  $$
  \Gamma_s^I=\bigl\{(\alpha,\beta)\in\Alpha_s^I\times\Alpha_s^I:\forall
  r\sum_c\alpha_{rc}=\sum_c\beta_{rc}\ \&\ \forall
  c\sum_r\alpha_{rc}=\sum_r\beta_{rc}\bigr\}.
  $$
  The set $\PP_s^I$ of connected $p$-gonal 
relations\index{connected $p$-gonal relation} in $I$ is the
  subset of those $(\alpha,\beta)\in\Gamma_s^I$ that cannot be
  decomposed into two other nonempty polygonal relations without row
  nor column in common:
%whenever
%$(\alpha,\beta)=(\alpha^1,\beta^1)+(\alpha^2,\beta^2)$ with
%$(\alpha^i,\beta^i)\in\Gamma_{s_i}^I$, $s_i\ge1$, $s=s_1+s_2$, then
%there is a row $r$ such that
%$\sum_c\alpha^1_{rc},\sum_c\alpha^2_{rc}\ge1$ or a column $c$ such
%that $\sum_r\alpha^1_{rc},\sum_r\alpha^2_{rc}\ge1$.
%
  $$
  \left\{\begin{array}{l}
      (\alpha,\beta)=(\alpha^1,\beta^1)+(\alpha^2,\beta^2)\\
      \hbox{with }(\alpha^i,\beta^i)\in\Gamma_{s_i}^I\ \&\ s_i\ge1
\end{array}\right.
\quad\Imp\qquad \left|\begin{array}{l}
    \exists r,c_1,c_2\quad\alpha^1_{rc_1},\alpha^2_{rc_2}\ge1\hbox{ or}\\
    \exists c,r_1,r_2\quad\alpha^1_{r_1c},\alpha^2_{r_2c}\ge1
\end{array}\right.
$$
\end{dfn}
%
\exa
$(i)$ for an octagon in
$\{0,1\}\times\{0,1\}\cup\{1,2\}\times\{2,3\}$:
$$
(0,0)\to(0,1)\downto(1,1)\to(1,3)\downto
(2,3)\to(2,2)\downto(1,2)\to(1,0)\downto.
$$
%$[\![0,3]\!]\times[\![0,3]\!]$. 
%Note that a polygon is equivalent to all its cyclic permutations of
%even order 

$(ii)$ $(\alpha=\e_{00}+\e_{11}+\e_{22}+\e_{33},
\beta=\e_{01}+\e_{10}+\e_{23}+\e_{32})$ is an octagonal relation that
is not connected: consider $\alpha^1=\e_{00}+\e_{11}$,
$\beta^1=\e_{01}+\e_{10}$, $\alpha^2=\e_{22}+\e_{33}$,
$\beta^2=\e_{23}+\e_{32}$.

\vskipa
The next proposition shows that, for our purpose, connected $p$-gonal
relations describe completely $p$-gons.
\begin{prp}\label{gon:bijection}
%
  Let $p=2s\ge0$ be an even integer and \DI/.  The mapping
%that maps 
  $P=((r_1,c_1),(r_1,c_2),\dots,(r_s,c_1))\mapsto(\alpha,\beta)$,
  where
  $$
%(r_1,c_1)\to%
%(r_1,c_2),\dots(r_s,c_1))\mapsto
%\bigl((\mes{\{i:(r_i,c_i)=q\}})_{q\in I},
%     (\mes{\{i:(r_i,c_{i+1})=q\}})_{q\in I}\bigr)
  \alpha_q=\mes{\{i:(r_i,c_i)=q\}}\quad \&\quad
  \beta_q=\mes{\{i:(r_i,c_{i+1})=q\}},
  $$
  is a surjection of the set of
% equivalence classes of 
  $p$-gons in $I$ onto the set $\PP_s^I$ of connected $p$-gonal
  relations in $I$. We shall write $P\sim (\alpha,\beta)$ and call
  $\gamma_{\alpha\beta}$ the number
%$\mes{G^{-1}(\alpha,\beta)}$ 
  of
%equivalent 
  $p$-gons mapped on $(\alpha,\beta)$.
%
\end{prp}
%
\dem
%Indeed, the image of a $p$-gon depends only upon its equivalence
%class. Further $G$ is injective. 
%Let us show that $G$ is
%surjective. 
Let $(\alpha,\beta)\in\PP_s^I$. Consider a rectangular line
$$(r_1,c_1)\to(r_1,c_2)\downto\dots\to(r_j,c_{j+1})\in I^{2j}$$ 
such that $\alpha^1_q=\mes{\{i:(r_i,c_i)=q\}}\le\alpha_q$ and
$\beta^1_q=\mes{\{i:(r_i,c_{i+1})=q\}}\le\beta_q$ and $j$ is maximal.
We claim $(a)$ that $c_{j+1}=c_1$ and $(b)$ that $j=s$. Let
$(\alpha^2,\beta^2)=(\alpha,\beta)-(\alpha^1,\beta^1)$.

\bm{a} If $c_{j+1}\ne c_1$, then
$\sum_r\alpha^2_{rc_{j+1}}=\sum_r\beta^2_{rc_{j+1}}+1\ge1$. Thus there
is $r_{j+1}$ such that $\alpha^2_{r_{j+1}c_{j+1}}\ge1$. But then
$\sum_c\beta^2_{r_{j+1}c}=\sum_c\alpha^2_{r_{j+1}c}\ge1$ and there is
$c_{j+2}$ such that $\beta^2_{r_{j+1}c_{j+2}}\ge1$: $j$ is not
maximal.

\bm{b} Suppose $j<s$. Then $(\alpha^2,\beta^2)\in\Gamma_{s-j}^I$ is
nonempty. As $(\alpha,\beta)$ is connected, there is $r,c,c'$ such
that $\alpha^1_{rc},\alpha^2_{rc'}\ge1$ or $r,r',c$ such that
$\alpha^1_{rc},\alpha^2_{r'c}\ge1$. Now our problem is invariant under
transposition; further it is invariant under the cyclic permutations
of even order
%
\begin{equation}\label{gon:cycle}
%
(r_k,c_k)\to(r_k,c_{k+1})\downto\dots
%\to(r_s,c_1)\downto
%(r_1,c_1)\to
%(r_1,c_2)\downto
%\dots
\to(r_{k-1},r_k)\downto.
%
\end{equation}
%
We may thus conclude without loss of generality that for $r'_1=r_j$
there is $c'_1$ such that $\alpha^2_{r'_1c'_1}\ge1$. Then there is
$c'_2$ such that $\beta^2_{r'_1c'_2}\ge1$. By the argument of $(a)$,
there is a $2j'$-gon in $(\alpha^2,\beta^2)$ of the form
$(r'_1,c'_1)\to(r'_1,c'_2)\downto\dots\to(r'_{j'},c'_1)\downto$.  Then
the following $2(j+j')$-gon
$$
(r_1,c_1)\to\dots\downto(r_j,c_j)\to(r'_1,c'_2)\downto
%(r'_2,c'_2)\to
\dots\to(r'_{j'},c'_1)\downto(r'_1,c'_1)\to(r_j,c_1)\downto
$$
shows that $j$ is not maximal.  \eck
%

\vskipa
We now introduce combinatorial properties in order to visualize the
special class of $p$-gons that we shall discover in the next section. 
\begin{dfn}\label{gon:dfn:track}
%
  Let $L$ be a closed rectangular line $q_1\to q_2\to\dots\to q_p\to
  q_1$.
%the $2s$-gon 
%$(r_1,c_1)\to(r_2,c_2)\to\dots\to(r_p,c_p)\to(r_1,c_1)$. 
  
  \bm{i} $L$ does not span a 
surface\index{p-gon@$p$-gon!that does not span a surface} 
if the index of every point in
  $\R\times\R$ is $0$ with respect to $L$: the bounded open set inside
  $L$ is empty.
  
  \bm{ii} $L$ returns on its 
track\index{p-gon@$p$-gon!that returns on its track}  if
%goes back the path it takes if 
  $L$ takes every elementary segment $[(r,c),(r,c+1)]$ and 
  $[(r,c),(r+1,c)]$ in $\N\times\N$ as many times in one sense as in
  the other sense.
%$$
%\forall (r,c)\in\N\times\N\  
%\N\left\{\begin{array}{l}
%\mes{\{j:r_j=r\ \&\ c_j\le c<c_{j+1}\}}=
%\mes{\{j:r_j=r\ \&\ c_{j+1}\le c<c_j\}}
%\mes{\{j:q\in[q_j,q_{j+1\[<q\}}=
%\mes{\{j:q\in[qr_j=r\ \&\ c_{j+1}\le c<c_j\}}
%$$
%$$
%\forall (r,c)\in\N\times\N\ 
%\mes{\{j:c_j=c\ \&\ r_{j-1}\le r<r_j\}}=
%\mes{\{j:c_j=c\ \&\ r_j\le r<r_{j-1}\}}
%$$
%\end{array}\right.
%
  
  \bm{iii} Two rectangular lines are 
similar\index{similar rectangular lines} 
if they are equivalent up
  to juxtaposed rectangular lines that return on their track.
 
%The double points of $P$ are those vertices of $P$ that
%  appear twice or more.
\end{dfn}
%
\begin{prp}\label{gon:dec}
%
  Let $P\sim(\alpha,\beta)$ be the $2s$-gon
  $(r_1,c_1)\to\dots\to(r_s,c_1)\downto$ and $q\in\N\times\N$.
  
  \bm{i} A closed rectangular line is similar to a polygon.
  
  \bm{ii} If $\alpha_q,\beta_q\ge1$, then $P$ is equivalent to the
  juxtaposition of a closed rectangular line containing $q$ that
  returns on its track and two disconnected polygons
  $P_1\sim(\alpha^1,\beta^1)$, $P_2\sim(\alpha^2,\beta^2)$ such that
  $(\alpha,\beta)=(\alpha^1,\beta^1)+(\alpha^2,\beta^2)+(\e_q,\e_q)$.
  
  \bm{iii} If $\alpha_q\ge2$ \(\vs $\beta_q\ge2$\), then $P$ is the
  juxtaposition of two nonempty polygons $P_1\sim(\alpha^1,\beta^1)$,
  $P_2\sim(\alpha^2,\beta^2)$ such that $\alpha^1_q,\alpha^2_q\ge1$ 
  \(\vs $\beta^1_q,\beta^2_q\ge1$\).
%
\end{prp}
%
\dem $(i)$ Note that two row segments or two column segments in
succession are similar to a single one.

$(ii)$ We may suppose $q=(r_1,c_1)$ by a cyclic permutation
\Ref{gon:cycle}. Let $1\le j\le s$ be such that $(r_j,c_{j+1})=q$.
Consider the two following polygons:
$$
P_1=(r_j,c_j)\to(r_1,c_2)\downto\dots\to(r_{j-1},c_j)\downto,
$$
$$
P_2=(r_{j+1},c_{j+1})\to\dots\downto(r_s,c_s)\to(r_s,c_1)\downto.
$$
$P$ is the juxtaposition of $(r_1,c_1)\to(r_1,c_2)$, $P_1$ deprived
of its first segment,
$(r_j,c_j)\to(r_j,c_{j+1})\downto(r_{j+1},c_{j+1})$, $P_2$ deprived of
its last segment and $(r_s,c_1)\downto(r_1,c_1)$.  Thus $P$ is
equivalent to the juxtaposition of $P_1$ and $P_2$, with in between

\bloc $(r_1,c_1)\to(r_1,c)\to(r_1,c_1)$, where $c$ is the point among
$c_1,c_2,c_j$ which is between the two others plus

\bloc $(r_1,c_1)\downto(r,c_1)\downto(r_1,c_1)$, where $r$ is the point among
$r_1,r_j,r_{j+1}$ which is between the two others.

If $P_1$ and $P_2$ are connected, then they may be glued as in the
proof of Proposition \ref{gon:bijection}$(b)$ and we set $P_1$ for
this glued polygon and $P_2$ for the empty polygon.

$(iii)$ As our problem is invariant under transposition and under the
cyclic permutations \Ref{gon:cycle}, we may suppose without loss of
generality that $\alpha_q\ge2$ and $q=(r_1,c_1)$. Let $j$ be such that
$(r_j,c_j)=q$. Set
$$
P_1=(r_1,c_1)\to\dots\to(r_{j-1},c_j)\downto\quad,\quad 
P_2=(r_j,c_j)\to\dots\to(r_s,c_1)\downto.\eqno{\ecks}
$$
%
\begin{cor}\label{gon:dec:cor}
%
  Let $P\sim(\alpha,\beta)$ be a rectangular polygon.
  
  \bm{i} $P$ is equivalent to the juxtaposition of closed rectangular
  lines that return on their track and disconnected polygons
  $P_j\sim(\alpha^j,\beta^j)$ such that $\alpha^j_q=0$ or
  $\beta^j_q=0$ for every $q\in\N\times\N$.
  
  \bm{ii} If $\alpha=\beta$, then $P$ is equivalent to the
  juxtaposition of connected closed rectangular lines that return on
  their track: $P$ is similar to the empty polygon.
  
  \bm{iii} $P$ is the juxtaposition of polygons
  $P_j\sim(\alpha^j,\beta^j)$ such that $\alpha^j_q,\beta^j_q\le1$
  for every $q\in\N\times\N$. 

%  $P$ is equivalent to the juxtaposition of closed
%  rectangular lines that return on their track and polygons that do
%  not loop.
%
\end{cor}
%
\dem $(i)$ Use Proposition \ref{gon:dec}$(ii)$ in a maximality argument 
and note
that closed curves contained in disconnected polygons are
disconnected.

$(ii)$ Note that if $\alpha=\beta$, then $\alpha^j=\beta^j$ in $(i)$: the $P_j$'s 
are disconnected. 

$(iii)$ Apply Proposition \ref{gon:dec}$(iii)$ in a 
maximality argument.\eck

\vskipa
We are now able to describe precisely the combinatorial properties
introduced in Definition \ref{gon:dfn:track}.







\begin{thm}\label{gon:thm}
\index{p-gon@$p$-gon!that returns on its track}
\index{p-gon@$p$-gon!that does not span a surface}
%
  Let $P\sim(\alpha,\beta)$ be the $2s$-gon
  $(r_1,c_1)\to\dots\to(r_s,c_1)\downto$. The following are
  equivalent.
  
  \bm{i} $\alpha=\beta$.
  
  \bm{ii} $P$ returns on its track.
  
  \bm{iii} $P$ does not span a surface.
  
  \bm{iv} For every vertex $q$ of $P$, there are as many row
  segments in $P$ reaching $q$ as leaving $q$.
  
  \bm{v} For every vertex $q$ of $P$, there are as many column
  segments in $P$ reaching $q$ as leaving $q$.

%\bm{iv}

\end{thm}
%
\dem $(i)\imp(ii)$ is Corollary \ref{gon:dec:cor}$(ii)$.
$(ii)\imp(iii)$ follows from the definition of the index as a path
integral.
%Let us prove this by induction on $s$. If $s=1$, then the $2$-gon
%$(r_1,c_1)\to(r_1,c_1)\downto$ trivially goes back the path it
%takes. Suppose 
%now that $(i)\imp(ii)$ is true for all $2j$-gons with $j<s$. 

%As $\alpha_{r_1c_1}=\beta_{r_1c_1}\ge1$, there is $1\le j\le s$ such
%that $(r_1,c_1)=(r_j,c_{j+1})$. Consider the two following polygons.
%% are
%%either empty ($j=1$ or $j=s$) or satisfy the induction hypothesis.
%$$
%P_1=(r_j,c_j)\to(r_1,c_2)\downto\dots\to(r_{j-1},c_j)\downto,
%$$
%$$
%P_2=(r_{j+1},c_{j+1})\to\dots\downto(r_s,c_s)\to(r_s,c_1)\downto.
%$$
%But then the path of $P$ consists of $(r_1,c_1)\to(r_1,c_2)$;
%the path of $P_1$ deprived of  its first segment
%$(r_j,c_j)\to(r_1,c_2)$; 
%$(r_j,c_j)\to(r_j,c_{j+1})\downto(r_{j+1},c_{j+1})$; the path of
%$P_2$ deprived of its last segment
%$(r_s,c_1)\downto(r_{j+1},c_{j+1})$;  
%$(r_s,c_1)\downto(r_1,c_1)$.

%Now our conclusion holds if and only if we insert the segments $(r_1,c_2)
%Now $P_1$ and $P_2$ are either empty ($j=1$ or $j=s$) or satisfy the
%induction hypothesis. It thus remains to show that the circuit 
%$$(r_1,c_1)\to(r_1,c_2)\to(r_j,c_j)\to(r_j,c_{j+1})\downto(r_{j+1},c_{j+1})\downto(r_s,c_1)\downto(r_1,c_1)$$
%goes back the path it takes, which is the case.

$(iii)\imp(iv)$ Let $q\in\N\times\N$. The number of row segments of
$P$ taking the elementary segment $[q,q+(1,0)]$ in the positive sense
minus those taking $[q,q+(1,0)]$ in the negative sense is exactly
twice the difference of the index of $q+(-1/2,1/2)$ with respect to
$P$ minus the index of $q+(1/2,1/2)$ with respect to $P$
\cite[VII(6.6)]{di68}. As these indices are equal, there are as many
row segments of $P$ taking $[q,q+(1,0)]$ in the positive as in the
negative sense. For the same reason, there are as many row segments of
$P$ taking $[q-(1,0),q]$ in the positive as in the negative sense.
Note further that row segments of $P$ that pass through $q$ take
necessarily $[q,q+(1,0)]$ and $[q-(1,0),q]$ in the same sense. Thus
the number of row segments reaching $q$ is equal to the number of row
segments leaving $q$ and $\alpha_q=\beta_q$.

$(iv)\imp(v)$ Because $P$ is a closed curve, there are as many
segments reaching $q$ as leaving $q$.

$(v)\imp(i)$ Let $q\in\N\times\N$. Note that $\alpha_q$ is exactly the
number of column segments reaching $q$ plus
$\mes{\{j:(r_{j-1},c_j)=(r_j,c_j)=q\}}$; $\beta_q$ is exactly the number
of column segments leaving $q$ plus
$\mes{\{j:(r_j,c_{j+1})=(r_{j+1},c_{j+1})=q\}}$.  \eck

%If $P$ goes back the path
%it takes, then there are as many vertical segments reaching $q$ as
%leaving $q$: indeed, the segments of $P$ that do not stop at $q$ reach
%$q$ from one sense and leave $q$ by the opposite sense. Thus
%$\alpha=\beta$.

\section{$1$-unconditional basic sequences of entries}

Let us recall the following definitions.
%
\begin{dfn}[\protect{\cite[\S4]{ha98}}]
%
  Let \DI/ and $p\ge1$.
  
  \bm{i} $I$ is 
%a \ES{p} set if $\{\e_q\}_{q\in I}$ is 
  an unconditional basic 
sequence\index{unconditional basic sequence of entries} 
in $S^p$ if for some $C$ 
  $$
  \sup_\pm\Bigl\|\sum_{q\in I}\pm a_q\e_q\Bigr\|_p\le
  C\Bigl\|\sum_{q\in I}a_q\e_q\Bigr\|_p.
  $$
  This amounts to the uniform boundedness of the family of
  relative Schur multipliers by signs
%
\begin{equation}\label{sigma:schur}
%
\Mu_\epsilon\colon S^p_I\to S^p_I,
(x_{rc})\mapsto(\epsilon_{rc}x_{rc})\quad\hbox{with }
|\epsilon_{rc}|=1.
%
\end{equation}
%
$I$ is complex \(\vs real\) 
$1$-unconditional\index{$1$-unconditional basic sequence of entries}
if all these multipliers
are isometries for $\epsilon_{rc}\in\T$ \(\vs
$\epsilon_{rc}\in\D$\). Then 
the norm of $x\in S^p_I$ does not depend on the complex \(\vs
real\) signs of its matrix coefficients. 

\bm{ii} $I$ is a 
c.b.\index{completely bounded unconditional basic sequence of entries}\ 
unconditional basic sequence if the family
\Ref{sigma:schur} is furthermore uniformly c.b. 
$I$ is a 
complex \(\vs real\) c.b.\ 
$1$-unconditional 
\index{completely bounded $1$-unconditional basic sequence of entries} 
basic sequence if the family
\Ref{sigma:schur} consists of c.b.\ isometries 
for $\epsilon_{rc}\in\T$ \(\vs $\epsilon_{rc}\in\D$\).
%
\end{dfn}
%
Notorious examples are single rows,  
%$\{(r,c):r=d\}$, 
single columns, 
%$\{(r,c):c=d\}$, 
single diagonals, 
% $\{(r,c):r-c=d\}$ 
single anti-diagonals and more generally sets $I$ such that for each
$(r,c)\in I$, there is no other element of $I$ in the row $r$ or in
the column $c$.
%$\{(r,c):r+c=d\}$.
%By the noncommutative Khinchin inequalities \cite{lu86,lp91}, $(i)$
%means exactly that 

\vskipa
We now undertake the matrix counterpart of the computation presented
in Section \ref{chap1}.\ref{ss:isom}
\begin{ldc}\label{sigma:ldc}
%
  Let $p=2s$ be an even positive integer and \DI/. Put
  $$
  \Phi_I(\epsilon,z)= \trace{\Bigl|\sum_{q\in
      I}\epsilon_qz_q\e_q\Bigr|}^p\quad \hbox{for
    }\epsilon\in\U^I\hbox{ and }z\in\C^{(I)}.
  $$
  Then
%
\begin{equation}\label{gon:expand}
%
\Phi_I(\epsilon,z)=\sum_{(\alpha,\beta)\in\PP_s^I}
\gamma_{\alpha\beta}\epsilon^{\beta-\alpha}\overline{z}^\alpha z^\beta
%
\end{equation}
%
where $\gamma_{\alpha\beta}$ is a positive integer for every
$(\alpha,\beta)\in\PP_s^I$. 
%defined in Proposition \ref{gon:bijection}.
%
\end{ldc}
%
\dem Let us expand $\Phi_I$.
%
\begin{eqnarray}
%
\nonumber\Phi_I(\epsilon,z)&=&\trace\Bigl(\sum_{(r,c),(r',c')\in I}
\epsilon_{rc}^{-1}\overline{z_{rc}}\e_{cr}
\epsilon_{r'c'}z_{r'c'}\e_{r'c'}
\Bigr)^s\\
\nonumber&=&\trace\sum_
{\scriptstyle(r_1,c_1),(r'_1,c'_1),\dots,\atop
\scriptstyle(r_s,c_s),(r'_s,c'_s)\in I}
\prod_{i=1}^s
\epsilon_{r_ic_i}^{-1}\overline{z_{r_ic_i}}\e_{c_ir_i}
\epsilon_{r'_ic'_i}z_{r'_ic'_i}\e_{r'_ic'_i}\\
&=&\sum_
{\scriptstyle(r_1,c_1),(r_1,c_2),\dots,\atop
\scriptstyle(r_s,c_s),(r_s,c_1)\in I}
\prod_{i=1}^s\epsilon_{r_ic_i}^{-1}\epsilon_{r_ic_{i+1}}
\overline{z_{r_ic_i}}z_{r_ic_{i+1}}\label{sigma:COMP}
%
\end{eqnarray}
%
with the convention $c_{s+1}=c_1$. 
Thus this sum runs over all $p$-gons in $I$. As the summand is equal
for $p$-gons that are associated to the same $p$-gonal relation
$(\alpha,\beta)$, Proposition \ref{gon:bijection} yields
\Ref{gon:expand}.  \eck
%

\vskipa
The following definition shows up in the analysis of the above
computation. 
\begin{dfn}
%
  Let \DI/ and $s\ge1$. $I$ is matrix 
  $s$-independent\index{matrix independent set of entries} if, given
  $q,q'\in I$, all rectangular lines of $s$ or less segments from $q$
  to $q'$ are similar: in other words, 
  there is only one rectangular line from $q$ to
  $q'$ up to rectangular lines that return on their track.
%
\end{dfn}
%
Computational lemma \ref{sigma:ldc} and Theorem
\ref{gon:thm} yield now the main theorem of this chapter.
\begin{thm}\label{sigma:THM}
\index{$1$-unconditional basic sequence of entries}
%
  Let \DI/ and $p=2s$ a positive even integer. The following 
  assertions are equivalent.
  
  \bm{i} $I$ is a c.b.\ complex $1$-unconditional basic sequence in
  $S^p$. 

  \bm{ii} $I$ is a complex $1$-unconditional basic sequence in $S^p$.
  
  \bm{iii} $I$ is a real\index{real vs.\ complex}\index{complex vs.\ real}
  $1$-unconditional basic sequence in $S^p$.
  
  \bm{iv} every $2s'$-gon $P$ in $I$ with $s'\le s$ satisfies the
  equivalent properties $(i)$--$(v)$ in Theorem \ref{gon:thm}.

  \bm{v} If $s$ is even, all rectangular lines of $s-1$ or less
  segments between two given rows, or equivalently between two
  given columns, are similar. If $s$ is odd, all rectangular lines
  of $s-1$ or less segments between a given row and a given column are
  similar.
  
  \bm{vi} $I$ is matrix $s$-independent.
%  
%  \bm{iv} every $2s$-gon $P$ in $I$ satisfies the equivalent
%  properties $(i)$--$(v)$ in Theorem \ref{gon:thm}.
%  \bm{v} There is no simple $t$-gon with $t\le s$ in $I$.
  
%  \bm{vi} No polygon of $2s$ vertices or less in $I$ encloses a
%  surface.
  
%  \bm{vii} Every polygon of $2s$ vertices or less goes back the path
%  it takes.
%
\end{thm}
%
\dem $(i)\imp(ii)\imp(iii)$ is trivial. 

%Let $C_1$ and $C_2$ be two paths from $q_1$ to $q_2$.  Let $q\to
%q_1$ and $q\to q_2$ be the first segments of $C_1$ and $C_2$. If $q_1$
%and $q_2$ are in the same row or column, let us modify $C_1$ by
%removing its first segment and $C_2$ by replacing its first segment by
%$q_1\to q_2$. Let $q'_1\to q'$ and $q'_2\to q'$ be the last segments
%of $C_1$ and $C_2$. If $q'_1$ and $q'_2$ are in the same row or
%column, let us modify $C_1$ by replacing its last segment by $q'_1\to
%q'_2$ and $C_2$ by removing its last segment.  Let now $P$ be the
%juxtaposition of $C_1$ and $-C_2$. $P$ is a polygon and ...
$(iii)\imp(iv)$ If $I$ is a real $1$-\ubs/ and $P\sim(\alpha,\beta)$
is a $2s$-gon in $I$, then $\alpha\equiv\beta$ mod.\ $2$ by
Computational lemma \ref{sigma:ldc}. Take now a $2s'$-gon
$P\sim(\alpha,\beta)$ in $I$. Suppose first that
$\alpha_q,\beta_q\le1$ for all $q\in I$. Let $q_1$ be the first vertex
of $P$ and $P'$ be the juxtaposition of the $2(s-s')$-gon $q_1\to
q_1\downto\dots\to q_1$ on $P$. Then
$P'\sim(\alpha,\beta)+(s-s')(\e_q,\e_q)$ and, by hypothesis,
$\alpha+(s-s')\e_q\equiv\beta+(s-s')\e_q$ mod.\ $2$ and
$\alpha=\beta$. Suppose now that $P$ is a general $2s'$-gon
$P\sim(\alpha,\beta)$ in $I$. Then, by Corollary 
\ref{gon:dec:cor}$(iii)$,
$P$ is the juxtaposition of polygons $P_j\sim(\alpha^j,\beta^j)$ such
that $\alpha^j_q,\beta^j_q\le1$ for every $q\in\N\times\N$. But then
$\alpha^j=\beta^j$ for each $j$ and thus $\alpha=\beta$. 

$(iv)\imp(v)$ We shall only treat the case of an even $s$ and two
given columns $s$ and $s'$. Take two rectangular lines $L,L'$ from
column $c$ to column $c'$: we may suppose that none of their points
but the first ones are on the column $c$; none of their points
but the last ones are on the column $c'$.
%We may suppose that their first and last
%segments are row segments; 
%we may furthermore suppose that row
%segments are followed by column segments and vice versa:
%$$
%L=(r_1,c)\to(r_1,c_2)\to\dots\to(r_j,c_j)\to(r_j,c').
%$$
%$$
%L'=(r'_1,c)\to(r'_1,c'_2)\to\dots\to
%(r'_{j'},c'_{j'})\to(r'_{j'},c').
%$$
The juxtaposition of $L$, the column segment from the last point of
$L$ to the last point of $L'$, 
%$(r_j,c')\downto(r'_{j'},c')$, 
$-L'$ and the column segment from the first point of $L'$ to the first
point of $L$ is similar to a polygon $P$ of $2s$ or less vertices and
thus by $(iv)$ 
to the empty polygon. Then the first and last points of $L$ and
$L'$ must be equal and $L$ and $L'$ are similar. 

$(v)\imp(vi)$ is trivial.

$(vi)\imp(iv)$ 
Let $P$ be a $2s'$-gon. Let $q$ be the first point of $P$ and $q'$ its
$(s'+1)$-st point. Then the $s'$ first
sides of $P$ and the $s'$ last sides taken in the negative direction
are two rectangular lines from $q$ to $q'$. They are similar and $P$
is similar to the empty polygon.

$(iv)\imp(i)$ holds by Computational lemma \ref{sigma:ldc} and Theorem
\ref{gon:thm}: in fact, the computation in the Computational lemma holds also
if the $z_{rc}$ are chosen operators in $S^p$ instead of complex
scalars, and if every $(\alpha,\beta)\in P_s^I$ satisfies
$\alpha=\beta$, then \Ref{sigma:COMP} is constant in $\epsilon$.\eck

\vskipa\rem
Harcharras\index{Harcharras, Asma} \cite{ha98} used 
Peller's\index{Peller, Vladimir V.} discovery \cite{pe82a} of the
link between Fourier and Hankel Schur 
multipliers\index{Hankel Schur multiplier} to produce
unconditional basic sequences in $S^p$ of the type
$\HM{E}=\{(r,c)\in\N\times\N:r+c\in E\}$, where $E\se\Z$. 
Such sets $\HM{E}$ are matrix $2$-independent if and only if $E$
is $2$-independent in the sense of Section \ref{chap1}.\ref{ss:isom}. 
But there are $3$-independent sets $E$ such that 
$\HM{E}$ is not matrix $3$-independent: consider
% for any $s\ge3$ if $\mes{E}\ge3$
$$
(n_1,0)\to(n_1,n_2)\downto(0,n_2)\to(0,n_1)\downto(n_2,n_1)\to(n_2,0)\downto
$$
with $n_3=n_1+n_2$, $n_2>n_1>0$ and $4n_2\ne3n_3$.
%In order to produce examples for matrix independent $I$, let us
%use Peller's discovery \cite{pe82a} of the link between Fourier and
%Hankel Schur multipliers.
%%
%\begin{prp}
%%
%  Let \DE/. $\HM{E}=\{(r,c)\in\N\times\N:r+c\in E\}$ is matrix
%  $s$-independent if and only if $E$ is $s$-independent \cite{ne98}.
%%
%\end{prp}
%%
%\dem
%Consider a $2s$-gon $P=(r_1,c_1)\to\dots\to(r_s,c_1)$ in $\HM{E}$: there
%are $n_1,\dots,n_{2s}\in E$ such that $r_j+c_j=n_{2j-1}$ and
%$r_j+c_{j+1}=n_{2j}$ ($1\le j\le\le s$). Then 
%$$
%-(r_1+c_1)+(r_1+c_2)-\dots+(r_s+c_1)=0=-n_1+n_2-\dots+n_{2s}.
%$$
%Thus every such $P$ in $I$ returns on its track if and only if
%$$\{n_1,n_3,\dots,n_{2s-1}\}=\{n_2,n_4,\dots,n_{2s}\}$$
%as soon as
%$n_1,\dots,n_{2s}\in E$ satisfy
%$n_1+n_3+\dots+n_{2s-1}=n_2+n_4+\dots+n_{2s}$.\eck











%
%\begin{thm}
%
%Let $p$ be an even positive integer and \DI/. Then $I$ is a real, and
%at the same times complex, $1$-unconditional basic sequence in $S^p$
%if and only if $I$ does not contain any $p$-gon, if and only if $I$
%does not contain any simple $r$-gon for even $2\le r\le p$.
%
%\end{thm}
  

%
%
%
%\baselineskip=11pt 
%\bibliography{references}
%
%
%
%
%
%
%\end{document}

%\section{Three remarks on Harcharras' paper}


\section{Two remarks on a paper by Harcharras}

\subsection{The c.b.\ unconditionality constant of sum sets}



We generalize Harcharras'\index{Harcharras, Asma} \cite[Prop.\
2.8]{ha98}. 
\index{completely bounded unconditionality constant!of sum sets}
%
\begin{prp}
  Let $A,B\se\Z$ with $\mes{A}=\mes{B}=n$. Then, for any $p\ge1$, the
  c.b.\ unconditionality constant of $A+B$ in $\SLP{}$ is at least $\lfloor
  n^{1/3}\rfloor^{|1/2-1/p|}$.
\end{prp}
%
\dem We shall use an inductive construction of sets $A_i\se A$ and
$B_i\se B$ such that $\mes{A_i}=\mes{B_i}=i$ and
\begin{equation}
  \label{three:indep}
\forall a,a'\in A_i\ \forall b,b'\in B_i\quad
a+b=a'+b'\ \imp\ a=a'\hbox{ and }b=b'.
\end{equation}
%
\bloc Put $a_1=\min A$ and $b_1=\min B$.

\bloc Assume $A_i$ and $B_i$ are constructed. Put
$$
a_{i+1}=\min\{a\in A\setminus A_i:\forall a'\in A_i\ \forall b\ne
b'\in B_i\ a+b\ne a'+b'\}
$$
if such an element exists; else $A\setminus A_i\se
A_i+(B_i-B_i)\setminus\{0\}$ and $n-i\le i^2(i-1)$. If we are able to
construct $A_{i+1}=A_i\cup\{a_{i+1}\}$, put
$$
b_{i+1}=\min\{b\in B\setminus B_i:\forall b'\in B_i\ \forall a\ne
a'\in A_{i+1}\ a+b\ne a'+b'\}
$$
if such an element exists; else $B\setminus B_i\se
B_i+(A_{i+1}-A_{i+1})\setminus\{0\}$ and $n-i\le i^2(i+1)$.

\bloc We conclude from this construction that there are $A_i=\{\lst
ai\}\se A$ and $B_i=\{\lst bi\}\se B$ of cardinality the minimal $i$
such that $n\le
i^3+i^2+i$, and \af with $i=\lfloor n^{1/3}\rfloor$.

The end of the proof is the same as Harcharras'.
%It is well known that t
The unconditionality constant of the canonical basis of $S_p^i$ is
$i^{|1/2-1/p|}$: there is a Schur multiplier $\Mu:S_p^i\to S_p^i$,
$(x_{kl})\mapsto(\epsilon_{kl}x_{kl})$, such that $|\epsilon_{kl}|=1$
and $\|\Mu\|= i^{|1/2-1/p|}$. Put
$$
f_x=(\e_{a_k+b_l}x_{kl})= \left[
%\begin{array}{ccc}\e_{a_1}&&0\\&\ddots&\\0&&\e_{a_i}\end{array}
\matrix{\e_{a_1}&&0\cr
&\ddots&\cr
0&&\e_{a_i}\cr}
\right]x
\left[\matrix{\e_{b_1}&&0\cr&\ddots&\cr0&&\e_{b_i}\cr}
\right]\in {\fam0 L}^p_{A_i+B_i}(S_p^i).
$$
Then $\|f_x(t)\|_{S_p^i}=\|x\|_{S_p^i}$ for each $t\in\T$ and thus
$\|f_x\|_{{\fam0 L}^p(S_p^i)}=\|x\|_{S_p^i}$. Consider 
$\nu:A_i+B_i\to\T$ be such that
$\nu(a_k+b_l)=\epsilon_{kl}$. By
\Ref{three:indep}, $\nu$ is well defined. Let $\Nu$ be the operator of
convolution with $\sum_{j\in A_i+B_i}\nu(j)\e_j$ acting on $\SLP{A_i+B_i}$. As
$$
\Nu\otimes\Id_{S_p^i}(f_x)=(\e_{a_k+b_l}\epsilon_{kl}x_{kl})=
f_{\Mu x},
$$
$\|\Nu\otimes\Id_{S_p^i}(f_x)\|=\|\Mu x\|$. Now the c.b.\ 
unconditionality constant of $A_i+B_i$ is at least
$\|\Nu\otimes\Id_{S_p^i}\|\ge\|\Mu\|$.  \eck
%
\begin{cor}
%
  If \DE/ contains the sum of two infinite sets, then $E$ is not a
  c.b.\ \EL{p}\index{completely bounded \EL{p} set} set for any $p>2$.
%
\end{cor}
%
\exa $E=\{2^i-2^j:i>j\}$ is not a c.b.\ \EL{p} set for any $p>2$.
Indeed, $\{2^i-2^j\}=E\cup-E$ does not and if $E$ did, then also $-E$
and $E\cup-E$.
%{ha98}\supseteq\{2^i:i\hbox{ odd}\}+\{2^j:j\hbox{ even}\}$
%is not a c.b.\ \EL{p} set for any $p>2$.
  


\subsection{Two new sufficient conditions for \EL{2s} sets}


The proof of \cite[Th.\ 1.14]{ha98} and especially \cite[Prop.\ 
1.14]{ha98} contain implicitly the two following new means to
construct \EL{p} sets.
%
\begin{prp}\index{Lambda(p) set@\EL{p} set!sufficient conditions}
  Let \DE/ and $s\ge2$ and integer. Let
  $$
  r'_s(E,n)=\mes{\{q_1<\dots<q_s:q_1+q_2+\dots+q_s=n\}};
  $$
  $$
  z_s(E,n)=\mes{\{q\in E^s:q_i\hbox{ distinct \& }
    -q_1+q_2-\dots+(-1)^sq_s=n\}}.
  $$
  $E$ is a \EL{2s} set if it is a finite union of sets $E_j$ such
  that either $r'_s(E_j,n)$ or $z_s(E_j,n)$ is a bounded function of
  $n$.
\end{prp}
%
The number $r'_s$ simplifies Rudin's\index{Rudin, Walter}
\cite[1.6(b)]{ru60} number $r_s$ in 
that it considers only distinct $q_i$. This is very useful in
applications. The number $z_s$ is \cite[Def.\ 1.11]{ha98}; Harcharras
proves that if $z_s(E,n)$ is a bounded function of $n$, then $E$ is
even a c.b.\ \EL{p} set. Nevertheless we wish to point out that the
condition is new even for usual \EL{p} sets.

%Harcharras notes that in particular $E=\{2^i+2^j\}_{i,j\ge1}$ is a set
%such that $r'_s(E,n)$ is uniformly bounded while $z_s(E,n)$ is not.
%Let us show that the opposite may also happen. [...]
%

\baselineskip=15.2pt minus 1.2pt
\chapter{Random constructions inside lacunary sets}\label{chap3}


\def\set{E}
\def\element{n}
\def\SCL{\SC{\set}}
\def\SLL#1{\SL{#1}{\set}}
\def\SLPL{\SLL{p}}
\def\DL/{$\set\se\Z$}
\def\DLL/{$\set=\{\element_k\}\se\Z$}
\def\DLLL/{$\set=\{\element_k\}_{k\ge1}\se\Z$}




\section{Introduction}
%
%
%
%
%
The study of lacunary sets in Fourier analysis still suffers from a
severe lack of examples, in particular for the purpose of
distinguishing two properties. In order to bypass the individual
complexity of integer sets, one frequently resorts to random
constructions. In particular, Li \cite{li98} uses in his argumentation
a construction due to Katznelson \cite{ka73}  
%and a theorem of Bourgain \cite[Prop.\ 8.2(i)]{bo88} 
to discriminate the following two functional properties of certain
subsets \DL/: 

\vskipa\bloc
A Lebesgue integrable function on the circle with Fourier frequencies in 
$\set$ is in fact $p$-integrable for all $p<\infty$. This means that  
all spaces $\SLPL$ coincide for $p<\infty$, 
\ie $\set$ is a \EL{p}\index{Lambda(p) set@\EL{p} set} set for all
$p$ in Rudin's terminology. No sequence of polynomial growth has this
property \cite[Th.\ 3.5]{ru60}. By
Theorem \ref{lambda:katz-gen}, almost every sequence of a given
superpolynomial order of growth is \EL{p} for all $p$.

\vskipa\bloc
A bounded measurable function on the circle with Fourier frequencies in
$\set$ is in fact continuous up to a set of measure $0$. 
This means that $\SLL{\infty}$ and $\SCL$ coincide: $\set$ is a
\ER/\index{Rosenthal set} 
set. Every sequence of exponential growth is a Sidon set and therefore
has this property. By Bourgain's Theorem \ref{def:bourgain}, 
%\cite[Prop.\ 8.2(i)]{bo88}
almost every sequence
of a given subexponential order of growth fails the \ER/
property.\vskipb  

A Rosenthal set may contain arbitrarily
large intervals \cite{ro67} und thus fail the \EL{p} property. This
shows that these two properties cannot be characterized by some
order of growth, whereas the random method is so imprecise that it 
ignores a range of exceptional sets.  
On the other hand, Li shows that some set is \EL{p} for all $p$ and
fails Rosenthal: his 
construction witnesses for the quantitative overlap between
superpolynomial and subexponential order of growth.\vskipb

We come back to Li \cite{li98} for two reasons: in the first place, 
we have been unable to locate a published proof of Katznelson's
statement. We
provide one for a 
stronger statement in Section \ref{lambda}. In the second place,  we
want to 
precise and supple the random construction in the following sense: can
one distinguish the \EL{p} property and the Rosenthal property among
subsets of a 
certain given set~? That sort of questions has been investigated
by Bourgain\index{Bourgain, Jean} in \cite{bo89a}. We give the
following answer (see Th.\ \ref{def:bourgain-gen}): 

\vskipa
{\bf Main Theorem } 
{\it Consider a polynomial\index{polynomial sequences} 
sequence of integers, or the sequence of
primes\index{prime numbers}. Then some subsequence of it is \EL{p} for
all $p$ and at the 
same time fails the Rosenthal property.}
\vskipb

This is a special case of the more general question: does every set that
fails the Rosenthal property contain a subset that is \EL{p} for all
$p$ and still fails the Rosenthal property~? We should emphasize at
this point that neither of these notions has an arithmetic
description. In fact, the family  of Rosenthal sets is coanalytic non
Borel \cite{go91} and any description would be at least as complex
as their definition. This is why we study instead the following two
properties for certain subsets \DL/.

\vskipa\bloc
Any integer $n$ has at most one representation as the sum of $s$
elements of $\set$. This implies that $\set$ is \EL{2s} by
\cite[Th.\ 4.5(b)]{ru60}. 

\vskipa\bloc
$\set$ is \ud/ in Hermann Weyl's\index{Weyl, Hermann} sense: save for
$t\equiv0$ mod $2\pi$, 
the 
successive means of $\{\e^{\ii\element t}\}_{\element\in\set}$ tend to
$0$, which is the mean of $\e^{\ii t}$ over $[0,2\pi\mathclose[$. This
implies that $\set$ is not \ER/ by \cite[Lemma 4]{lu81}. \vskipb

Our random construction gives no hint for explicit procedures to build
such integer sets. The question whether some ``natural'' set of
integers is \EL{p} for all $p$ and fails the Rosenthal property
remains open. 

Let us describe the paper briefly. Section \ref{def} introduces the
inquired notions and gives a survey of former and new
results. As the right framework for this study appears to consist in
the sequences of polynomial growth, we give them a precise meaning in
Section \ref{poly}, and show that they are nicely distributed among
the intervals of the partition of \Z\ defined by
$\{\pm2^{k!}\}$. Section \ref{lambda} establishes an
optimal criterion for the generic subset of a set with polynomial
growth to be \EL{p} for all $p$. Section \ref{ergo} comes back to
Bourgain's proof in \cite[Prop.\ 8.2(i)]{bo88}: we simplify and
strengthen it in order to investigate the generic subset of an \ud/
set.

\vskipa{\bf Notation }
$\T=\{t\in\C:\,|t| = 1\}$ is the unit circle endowed with its Haar
measure $dm$ and \Z\ its dual group of
integers: for each $n\in\Z$, let $\e_n(t)=t^n$. The cardinal of \DLL/
is written \mes{\set}. We denote by $c_0(\T)$ the space of functions on \T\
which are arbitrarily small outside finite sets; such functions
necessarily have countable support.

For a space $X$ of integrable functions on \T\ and \DL/, $X_\set$
denotes the 
space of functions with Fourier spectrum in $\set$: 
$X_\set=\bigl\{f\in X: \widehat{f}(n)=\int\e_{-n}f\,dm=0$ if $n\notin\set\bigr\}$.

We shall stick to Hardy's notation: $u_n\preccurlyeq v_n$ (\vs $u_n\ll
v_n$) if $u_n/v_n$ is bounded (\vs vanishes) at infinity.
%
%
%
%
%
%
\section{Equidistributed and \EL{p} sets}\label{def}
%
%
%
%
%
%
\begin{dfn}
%
Let \DLLL/ ordered by increasing absolute value $|\element_k|$. 

\bm{i}
\cite[Def.\ 1.5]{ru60}
Let $p>0$. $\set$ is a \EL{p} set if, for some ---~or equivalently for any~---
$0<r<p$, $\SLPL$ and $\SLL{r}$ coincide:
$$
\exists C_r\quad\forall f\in\SLPL\quad\|f\|_r\le\|f\|_p\le C_r\|f\|_r.
$$

\bm{ii} 
\cite[\S 7]{we16}
$\set$ is \ud/\index{equidistributed set} if for each $t\in\T\setminus\{1\}$ the
successive means 
%
\begin{equation}\label{def:def}
f_k(t)=\frac1k\sum_{j=1}^k\e_{\element_j}(t)\ \tol_{k\to\infty}\ 0.
%\hbox{\fam0\bf 1}_{\{1\}}(t)=\left\{\begin{array}{l}
%      \mbox{$1$ if $t=1$}\\
%      \mbox{$0$ if not.}
%\end{array}\right.
%
\end{equation}
%
Thus $E$ is equidistributed if and only if the sequence of characters
in $E$ converges to $\hbox{\fam0\bf 1}_{\{1\}}$ for the Ces\`aro
summing method. 
If $f_k$ tends pointwise to $f\in c_0(\T)$, then $\set$ is
\wud/\index{weakly equidistributed set}.
%
\end{dfn}
%
If $\set$ is \wud/, then $f$ defines an element of
$\SCL^{\perp\perp}$.  By Lust-Piquard's \cite[Lemma 4]{lu81}, $\SCL$
then contains a copy of $c_0$ and $\set$ cannot be Rosenthal. 
%See also \cite{lu89} for further background.

For example, \Z\ and \N\ are \ud/.  Arithmetic\index{arithmetic
  sequences} sequences are \wud/: there is a finite set on which
$f_k\nrightarrow0$. Polynomial sequences\index{polynomial sequences}
of integers (\cite[Th.\ 9]{we16} and \cite[Lemma 2.4]{va81}, 
see \cite[Ex.\ 2]{lu89}) 
and the sequence of prime\index{prime numbers} numbers 
(Vinogradov's\index{Vinogradov} theorem \cite{el75}, see \cite[Ex.\ 1]{lu89}) 
are \wud/: $f_k(t)$
may not converge to $0$ for rational $t$. There are nevertheless
sequences of bounded pace that are not \wud/ \cite[Th.\ 11]{et57}.
Sidon sets are \EL{p} for all $p$ \cite[Th.\ 3.1]{ru60}, but not \wud/
since they are Rosenthal. \vskipb

\exa
Consider the geometric\index{geometric sequences}
sequence $\set=\{3^k\}_{k\ge1}$ and the corresponding sequence of
successive means $f_k$. By
\cite[Th.\ 14]{et57}, the $f_k$ do not converge to 0 on a null set of
Hausdorff dimension $1$. Consider 
$$
f_k^j=k^{-j}\hskip-1em\sum_{1\le\lst kj\le k}\hskip-1.5em
\e_{3^{k_1}+\dots+3^{k_j}}=
k^{-j}\Bigl(\mathop{j!\sum}\limits_{1\le k_1<\dots<k_j\le k}+
\sum_
{\scriptstyle1\le\lst kj\le k\atop\hbox{\scriptsize not all distinct}}
\hskip-0.5ex\Bigr)
\e_{3^{k_1}+\dots+3^{k_j}}.
$$
Let $j\ge1$. Put 
$\set^{(j)}=\{3^{k_1}+\dots+3^{k_j}:\,1\le k_1<\dots<k_j\}$
and let $f_k^{(j)}$ be the corresponding successive means
\Ref{def:def}. Then
%
\begin{eqnarray*}
\|f_k^j-f_{(\!{k\atop j}\!)}^{(j)}\|_\infty&\le&
\Bigl(\Bigl({k\atop j}\Bigr)^{-1}-\frac{j!}{k^j}\Bigr)
\Bigl({k\atop j}\Bigr)+\frac1{k^j}
\Bigl(k^j-\frac{k!}{(k-j)!}\Bigr)\\
&=&2\Bigl(1-\frac{k!}{k^j(k-j)!}\Bigr)
\tol_k0.
\end{eqnarray*}
%
Thus $\set^{(j)}$, which is \EL{p} for all $p$ \cite[Th.\ IV.3]{me68}
and not Sidon, is not \wud/. \vskipb

However, as Li\index{Li, Daniel} notes, these two classes meet.
%
\begin{thm}[\cite{li98}]\label{def:li}
%
There is an \ud/ sequence that is \EL{p} for all $p$.
%
\end{thm}
%
{\it Sketch of proof.}\/ 
Li uses the following random construction, discovered by
Erd\H{o}s\index{Erd\H{o}s, Paul} \cite{er56,er60} and introduced to
harmonic analysis by 
Katznelson\index{Katznelson, Yitzhak} and 
Malliavin\index{Malliavin, Paul} \cite{km65,km66}.
%
\begin{construction}\label{def:constr}
\index{random construction}
%
Let \DL/ and consider independent $\{0,1\}$-valued selectors
$\xi_\element$ of mean $\delta_\element$ $(\element\in\set)$, \ie
$\P{\xi_\element=1}=\delta_\element$. Then the random set $\set'$ is
defined by
$$
\set'=\{\element\in\set:\ \xi_\element=1\}.
$$
%
\end{construction}
%
The first ingredient of the proof is Bourgain's\index{Bourgain, Jean}
following  
%
\begin{thm}[\protect{\cite[Prop.\ 8.2(i)]{bo88}}]\label{def:bourgain}
%\index{equidistributed set}
Let $\set=\N$ in Construction \ref{def:constr}. If
$\delta_\element$ decreases with $\element$ while
$\delta_\element\gg\element^{-1}$, then $\set'$ is almost surely \ud/.
%skip
\end{thm}
%
\rem
In this sense, almost every sequence of a subexponential 
growth\index{subexponential growth} 
given
by $\{\delta_\element\}$ is equidistributed: indeed, for almost every
$E'\se\N$, 
$$
\mes{\set'\cap[0,\element]}
\sim\smp\delta0\element\gg\log\element
$$
by the Law of Large Numbers. Note however that the set $\set^{(j)}$
defined in Example 2.2 has subexponential growth:
$\mes{\set^{(j)}\cap[-\element,\element]}\succcurlyeq(\log\element)^j$, and
is not \ud/.\vskipb



The second ingredient is a result announced without proof by
Katznelson.
%
\begin{prp}[\protect{\cite[\S 2]{ka73}}]\label{def:katz}
%
Put $I_k=\mathopen]p_{k-1},p_k ]$ with $p_k >p_{k-1}^2$ $(k\ge1)$. Let
$\set=\N$ in Construction \ref{def:constr}. There is a choice of
$(\ell_k)$ with $\ell_k\gg\log p_k$ such that for
$\delta_\element=\ell_k/\lmes I_k\rmes $ $(\element\in I_k)$, $\set'$
is \EL{p} for all $p$ almost surely.
%
\end{prp}
%
Li suggests to apply the content of Proposition \ref{def:katz} with
$p_k=2^k$ and $\ell_k=k$: then $\delta_\element\gg\element^{-1}$ and
Theorem \ref{def:li} derives from Theorem \ref{def:bourgain}.
\eck\vskipb

We shall generalize Katznelson's and Li's results with a new proof that
permits to construct $\set'$ inside of sets $\set$ with polynomial growth
(see Def.\ \ref{poly:def}) and yields an optimal criterion on
$\ell_k$. We shall subsequently generalize Theorem \ref{def:bourgain}
to obtain the Main Theorem via
%
\begin{thm}\label{def:bourgain-gen}
%
Let $\set$ be \ud/ \(\vs weakly\) and with polynomial
growth\index{polynomial growth}.
Then there is a subset $\set'\subseteq\set$ \ud/ \(\vs weakly\)
and at the same time \EL{p} for all $p$.
%
\end{thm}
%
See also Corollary \ref{ergo:COR} for a precise and quantitative statement.%
%
%
%
%
\section{Sets with polynomial growth}\label{poly}
%
%
%
%
%
We start with the definition and first property of such sets.
%
\begin{dfn}\label{poly:def}
%
Let \DLLL/ be an infinite set ordered by increasing absolute value and
$\set[t]=\lmes\set\cap[-t,t]\rmes $ its distribution 
function\index{distribution function}. 

\bm{i}
$\set$ has polynomial growth\index{polynomial growth} 
if $\element_k\preccurlyeq k^d$ for
some $1 \le d < \infty$. This amounts to $\set[t]\succcurlyeq t^\eps$
for $\eps=d^{-1}$.

\bm{ii}
$\set$ has regular polynomial growth\index{regular polynomial growth} 
if there is a $c>1$ such that 
$|\element_{\lceil ck\rceil}| \le 2|\element_k|$ for large $k$.
This amounts to $\set[2t] \ge c\set[t]$ for large $t$.
%
\end{dfn}
%
\dem
$(i)$
If $|\element_k| \le  C k^d$  for large $k$ and $C k^d \le  t < C (k+1)^d$,
then $\set[t] \ge k > (t/C)^\eps-1$. Conversely, if $\set[t] \ge c
t^\eps$ for large $t$ and $c (t-1)^\eps < k \le c t^\eps$, then
$|\element_k| \le  t < (k/c)^d+1$. 

$(ii)$ If $|\element_{\lceil ck\rceil}| \le  2|\element_k|$ for large $k$
and $k$ is maximal with $|\element_k|\le t$, then $\set[2t] \ge
\set[2|\element_k|] \ge c k = c \set[t]$. Conversely, if $\set[2t]
\ge c\set[t]$ for large $t$, then $\set[|\element_k|] \in \{k,k+1\}$ and 
$\set[2|\element_k|]\ge c k$. Thus $|\element_{\lceil ck\rceil}| \le
2|\element_k|$. 
\eck\vskipb

In particular, polynomial sequences\index{polynomial sequences} 
have regular polynomial growth. By
the Prime Number Theorem, the sequence of primes\index{prime numbers} 
also has.
Property $(ii)$ implies property $(i)$: if $\set[2t] \ge c\set[t]$ for
large $t$, then $\set[t]\succcurlyeq t^{\log_2c}$. The converse
however is false as shows
$F=\bigcup\, \mathopen]2^{2^{2k}},2^{2^{2k}+1}]$, for which $F[t]
\succcurlyeq t^{1/4}$ while $F[2t]=F[t]$ infinitely
often. Let us relate Definition \ref{poly:def} with certain partitions
of \Z. Regular growth means in fact that $\set$ is regularly distributed 
on the annular dyadic partition\index{dyadic partition} of \Z
%
\begin{equation}\label{poly:lpd}
%
{\cal P}=\bigl\{[-p_0,p_0],I_k=[-p_k,-p_{k-1}\mathclose[\cup
\mathopen]p_{k-1},p_k]\bigr\}_{k\ge1}\mbox{ where }p_k=2^k
%
\end{equation} 
%
and $F$ shows that there are sets with polynomial growth which are not
regularly distributed on the partition defined by
$p_k=2^{2^k}$. However, the intervals of the gross 
partition\index{gross partition} 
%
\begin{equation}\label{poly:lpg}
%
{\cal P}=\bigl\{[-p_0,p_0],I_k=[-p_k,-p_{k-1}\mathclose[
\cup\mathopen]p_{k-1},p_k]\bigr\}
\mbox{ where }\log p_k \gg\log p_{k-1}
%
\end{equation} 
%
grow with a speed that forces regularity. Put $p_k=2^{k!}$ for a
simple explicit example. We have precisely 
%
\begin{prp}\label{poly:regul}
%
Let \DL/, ${\cal P}=\{I_k\}$ a partition of \Z\ and $\set_k=\set\cap
I_k$. Then $\log\mes{\set_k}\succcurlyeq \log \mes{I_k}$ in the two
following cases:

\bm{i}
if $\set$ has regular polynomial growth and $\cal P$ is partition
$(\ref{poly:lpd})$;

\bm{ii}
if $\set$ has polynomial growth and $\cal P$ is partition
$(\ref{poly:lpg})$.
%
\end{prp}
%
\dem
$(i)$ Choose $K$ and $c>1$ such that $\set[2^k] \ge c\set[2^{k-1}]$
for $k\ge K$. Then $\set[2^k] \succcurlyeq c^k$.
%if \set is non-empty. 
Thus
$$
\mes{\set_k} = \set[2^k] - \set[2^{k-1}] \ge  
(1-c^{-1})\set[2^k] \succcurlyeq c^k=2^{k\log_2c}.
$$
$(ii)$
In this case $p_k ^\eps \gg p_{k-1}$ for any $\eps>0$. Now there is
$\eps>0$ such that 
$$
\mes{\set_k} = \set[p_k] - \set[p_{k-1}] \succcurlyeq p_k ^\eps \succcurlyeq
\mes{I_k}^\eps.\eqno{\eck}
$$
% 
%
%
%
%
\section{Sets that are \EL{p} for all $p$}\label{lambda}
%
%
%
%
%
In this section, we establish an improvement (Th.\
\ref{lambda:katz-gen}) of Katznelson's statement \cite[\S2]{ka73}. We
  first recall several known definitions and results. 

\EL{p} sets have a practical description in terms of
unconditionality. We shall also use a combinatorial property that is
more elementary than \cite[1.6(b)]{ru60}: to this end, write
$\Zeta_s^m$ for the following set of arithmetic relations.
$$
\Zeta_s^m=\bigl\{\zeta\in{\Z^*}^m:\,\sm\zeta m=0\mbox{ and }
|\zeta_1|+\dots+|\zeta_m|\le 2s\bigr\}. 
$$
Note that $\Zeta_s^1$ and $\Zeta_s^m$ $(m>2s)$ are empty, and that
every $\zeta\in\Zeta_s^2$  is of form $\zeta_1\cdot(1,-1)$: this is
the identity relation.
%
\begin{dfn}
%
Let $1\le p<\infty$, $s\ge1$ integer and \DL/.

\bm{i} \cite{ka48}
$\set$ is an unconditional basic sequence in $\SLP{}$ if 
$$
\sup_{\pm}\biggl\|\sum_{n\in\set}\pm a_\element\e_\element\biggr\|_p\le
C\biggl\|\sum_{\element\in \set}a_\element\e_\element\biggr\|_p.
$$
for some $C$. If $C=1$ works, $\set$ is a $1$-unconditional basic
sequence in $\SLP{}$.

\bm{ii}
[Chapter \ref{chap1}, \S\ref{ss:isom}] $\set$ is $s$-independent if $\sum\zeta_iq_i\ne0$ for all $3\le m\le 2s$, 
$\zeta\in\Zeta_s^m$ and distinct $\lst qm\in\set$.
%
\end{dfn}
%
\begin{prp}
%[cf.\ \protect{\cite[Prop.\ 2.5]{ne98}}]
\label{lambda:uncond}
%
Let $1\le p<\infty$, $s\ge1$ integer and \DL/.

\bm{i} 
\cite[proof of Th.\ 3.1]{ru60}
$\set$ is a \EL{\max(p,2)} set if and only if $\set$ is an
unconditional basic sequence in $\SLP{}$.

\bm{ii}
[Chapter \ref{chap1}, Prop.\ \ref{mub:isom}]
%\cite[Prop.\ 2.5]{ne98}
$\set$ is a $1$-unconditional basic sequence in $\SL{2s}{}$ if and only
if $\set$ is $s$-independent.
%
\end{prp}
%
We need to introduce a second classical notion of unconditionality
that rests on the Littlewood--Paley theory.
%
\begin{dfn}[\cite{hk89}]
%
Let ${\cal P }=\{I_k\}$ be a partition of \Z\ in finite
intervals. ${\cal P}$ is a Littlewood--Paley
partition\index{Littlewood--Paley partition} if for each
$1<p<\infty$ there is a constant $C_p$ such that 
%
\begin{equation}\label{lambda:paley}
%
\forall f\in\SLP{}\quad
\sup_{\pm}\Bigl\|\sum\pm f_k\Bigr\|_p\le C_p\|f\|_p\qquad
\mbox{with }\widehat{f_k}=
\left\{\begin{array}{l}\widehat{f}\mbox{ on}\\
0\mbox{ off}\end{array}\right. I_k. 
\end{equation}
%
\end{dfn}
%
By Khinchin's inequality, this means exactly that 
$$
\forall f\in\SLP{}\quad
\|f\|_p\approx\Bigl\|\Bigl(\sum|f_k|^2\Bigr)^{1/2}\Bigr\|_p.
$$
In particular, the dyadic partition $(\ref{poly:lpd})$ and the
gross partition $(\ref{poly:lpg})$ are Littlewood--Paley
\cite{lp31}. By Proposition \ref{lambda:uncond} and
\Ref{lambda:paley}, we obtain 
%
\begin{prp}\label{lambda:union}
%
Let $\{I_k\}$ be a Littlewood--Paley partition and $\set_k\subseteq
I_k$. If $\set_k$ is $s$-independent for each $k$, then
$\set=\bigcup\set_k$ is an unconditional basic sequence in $\SL{2s}{}$ and
thus a \EL{2s} set.
%
\end{prp}
%
We generalize now Katznelson's Proposition \ref{def:katz}.
%
%
\begin{lem}\label{lambda:estim}
%
Let $s\ge2$ integer, \DL/ finite and $0\le\ell\le\mes{\set}$. 
Put $\delta_n=l/\mes{E}$ in
Construction \ref{def:constr}, so that all selectors $\xi_\element$
have same distribution. Then there is a constant $C(s)$ that depends
only on $s$ such that
$$
\P{\set'\mbox{ is $s$-dependent\/}} \le 
C(s)\frac{\ell^{2s}}{\lmes \set\rmes }.
$$
%
\end{lem}
%
\dem
We wish to compute the probability that there are $3\le m\le 2s$,
$\zeta\in\Zeta_s^m$ and distinct $\lst qm\in\set'$ with
$\sum\zeta_iq_i=0$. As the number $C(s)$ of arithmetic relations 
$\zeta\in\Zeta_s^m$ ($3\le m\le 2s$) is finite and depends on $s$
only, it suffices to compute, for fixed $m$ and $\zeta\in\Zeta_s^m$
%
\begin{eqnarray*}
%
\lefteqn{\P{\exists\lst qm\in\set'\mbox{ distinct}:\,
\sum\zeta_iq_i=0}}\\
&=&\P{\exists\lst q{m-1}\in\set'\mbox{ distinct}:\,
-\zeta_m^{-1}\sum_{i=1}^{m-1}\zeta_iq_i\in
\set'\setminus\{\lst q{m-1}\}}\\
&=&\P{\bigcup_
{\scriptstyle\lst q{m-1}\atop
\scriptstyle\in\set'\mbox{\scriptsize\,distinct}}\hskip-1.5em
\Bigl[-\zeta_m^{-1}\sum_{i=1}^{m-1}\zeta_iq_i\in
\set'\setminus\{\lst q{m-1}\}\Bigr]}\\
&=&\P{\bigcup_
{\scriptstyle\lst q{m-1}\atop
\scriptstyle\in\set\mbox{\scriptsize\,distinct}}\hskip-1.5em
\Bigl[q_m=-\zeta_m^{-1}\sum_{i=1}^{m-1}\zeta_iq_i\in
\set\setminus\{q_i\}_{i=1}^{m-1}\ \&\ 
\xi_{q_1}=\dots=\xi_{q_m}=1\Bigr]}
%
\end{eqnarray*}
%
The union in the line above runs over 
$$
\frac{\lmes \set\rmes !}{(\lmes \set\rmes -m+1)!}\le\lmes \set\rmes ^{m-1}
$$
$(m-1)$-tuples. Further, the event in the inner brackets implies that
$m$ out of 
$\mes{\set}$ selectors $\xi_\element$ have value $1$: its probability is
bounded by $(\ell/\mes{\set})^m$. Thus
$$
\P{\set'\mbox{ is $s$-dependent}}\le
C(s)\max_{3\le m\le 2s}\lmes \set\rmes ^{m-1}\frac{\ell^m}{\lmes
\set \rmes ^m}\le
C(s)\frac{\ell^{2s}}{\lmes \set\rmes }.\eqno{\eck}
$$
%
The random method we shall use is the following random construction. 
%
\begin{construction}\label{lambda:constr}\index{random construction}
%
Let \DL/. Let $\{I_k\}$ be a Littlewood--Paley partition and
$\set_k=\set\cap I_k$. Let $(\ell_k)_{k\ge1}$ with $0\le\ell_k\le\lmes
\set_k\rmes $ and put 
$$
\P{\xi_\element=1}=\delta_\element=\ell_k/\mes{\set_k}\quad
(\element\in\set_k)
$$
in Construction \ref{def:constr}. Put $\set'_k=\set'\cap I_k$.
%
\end{construction}
%
\begin{thm}\label{lambda:katz-gen}
%
Let \DL/ have polynomial \(\vs regular\) growth and $\{I_k\}$ be the
gross \Ref{poly:lpg} \(\vs dyadic \Ref{poly:lpd}\)
Littlewood--Paley partition. Do Construction \ref{lambda:constr}. The
following assertions are equivalent.

\bm{i}
$\log \ell_k \ll \log \lmes I_k \rmes $, \ie $\log \ell_k \ll \log
p_k $ \(\vs $\log \ell_k \ll k$\);

\bm{ii}
$\set'$ is almost surely a \EL{p}\index{Lambda(p) set@\EL{p} set!for all $p$} 
set for all $p$. 
%
\end{thm}
%
\dem Note that by Proposition \ref{poly:regul}, there is a positive
$\alpha$ such that $\mes{\set_k}>\mes{I_k}^{\alpha}$ for large $k$. 
$(i)\imp(ii)$ Let $s\ge2$ be an arbitrary integer. By Proposition 
\ref{lambda:estim}, 
$$
\sum_{k=1}^\infty\P{\set'_k\mbox{ is $s$-dependent}}\le
C(s)\sum_{k=1}^\infty\frac{\ell_k^{2s}}{\lmes \set_k\rmes }.
$$
For each $\eta>0$, $\ell_k\le\mes{I_k}^\eta$ for large $k$. Choose
$\eta<\alpha/2s$. Then  $\ell_k^{2s}/\mes{\set_k}\le
\mes{I_k}^{2s\eta-\alpha}$  for large
$k$, and the series above converges since $\mes{I_k}\succcurlyeq
2^k$. By the Borel--Cantelli lemma, $\set'_k$ is almost surely
$s$-independent for large $k$. By Proposition \ref{lambda:union}, $\set'$ is
almost surely the union of a finite set and a \EL{2s} set. By
\cite[Th.\ 4.4(a)]{ru60}, $\set'$ itself is almost surely a \EL{2s} set.

$(ii)\imp(i)$ 
If $\set'$ is a \EL{2s} set, then by \cite[Th.\ 3.5]{ru60} or simply by
\cite[(1.12)]{bo89a}, there is a constant $C_s$ such that
$\mes{\set'_k}<C_s\mes{I_k}^{1/s}$. As $\mes{\set'_k}\sim\ell_k$ almost
surely by the Law of Large Numbers (cf.\ the following Lemma
\ref{ergo:bern}), one has $\log \ell_k \ll \log \mes{I_k}$.
\eck\vskipb

\rem
As one can easily construct sets that grow as slowly as one wishes and
nevertheless contain arbitrarily large intervals 
(see also \cite[Th.\ 3.8]{ru60}
for an optimal statement), one cannot remove
the adverb ``almost surely'' in Theorem
\ref{lambda:katz-gen}$(ii)$.\vskipb 

\rem
The right formulation of Katznelson's Proposition \ref{def:katz} 
thus turns out to be the following. Let $\set=\N$ and
$I_k=\mathopen]p_{k-1},p_k ]$ with $p_k >cp_{k-1}$ for some $c>1$ in
Construction \ref{lambda:constr}. Then $\set'$ is almost surely a \EL{p}
set for all $p$ if and only if $\log \ell_k \ll \log p_k$. \vskipb 

\rem
Theorem \ref{lambda:katz-gen} shows that there are sets that are
\EL{p} for all $p$ of any  given
superpolynomial\index{superpolynomial growth} order of growth. This is
optimal since sets with 
distribution $\set[t]\succcurlyeq t^\eps$ fail the \EL{p} property for
$p>2/\eps$ by \cite[Th.\ 3.5]{ru60}. Such sets may also be constructed
inductively by combinatorial means: see Section
\ref{chap1}.\ref{sect:comb} and \cite[\S II, (3.52)]{hr83}. 
%\cite[\S10]{ne98}.
%
%
%
%
%
\section{Equidistributed sets}\label{ergo}
%
%
%
%
%
In this section, we shall finally state and prove our principal
result. To this end, we shall first generalize Bourgain's Theorem
\ref{def:bourgain} in order to get 
Theorem \ref{ergo:bourgain-gen}. The following lemma is Bernstein's
distribution inequality\index{Bernstein, Serge} \cite{be64} and dates
back to 1924. 
%
%
\begin{lem}\label{ergo:bern}
%
Let $\lst Xn$ be complex independent random variables such that
%
\begin{equation}\label{ergo:bern:hyp}
|X_i|\le1\quad\mbox{and}\quad\E X_i=0\quad\mbox{and}\quad
\E|X_1|^2+\dots+\E|X_n|^2\le\sigma.
\end{equation}
%
Then, for all positive $a$,
%
\begin{equation}\label{ergo:bern:conc}
%
\P{|X_1+\dots+X_n| \ge a}<4\exp(-a^2/4(\sigma+a)).
%
\end{equation}
%
\end{lem}
%
\dem
%This is Bennett's variant of Bernstein's 
%probabilistic inequality \cite{be64}. 
Consider first the case of real random variables. By
\cite[(8b)]{be62}, 
$$
\P{X_1+\dots+X_n \ge a}<\exp(a-(\sigma+a)\log(1+a/\sigma));
$$
as $\log(1-u)\le-u-u^2/2$ for $0 \le u < 1$,
$$
\P{X_1+\dots+X_n \ge a}<\exp(-a^2/2(\sigma+a)).
$$
One gets \Ref{ergo:bern:conc} since for complex $z$
$$
|z| \ge a\quad\Longrightarrow\quad
\max(\Re z,-\Re z,\Im z,-\Im z) \ge a/\sqrt{2}.\eqno{\eck}
$$
%
The next lemma corresponds to \cite[Lemma 8.8]{bo88} and is the
crucial step in the estimation of the successive means of 
$\{\e^{\ii\element t}\}_{\element\in\set'}$. Note that its hypothesis
is not on the individual $\delta_\element$, but on their successive
sums $\sigma_k$: this is needed in order to cope with the irregularity
of $\set$.
%
\begin{lem}\label{ergo:bourgain-lem}
%
Let \DLL/ be ordered by increasing absolute value. Do Construction
\ref{def:constr} and put
$\sigma_k=\delta_{\element_1}+\dots+\delta_{\element_k}$. If $\sigma_k
\gg \log |\element_k|$, then almost surely
%
\begin{equation}\label{ergo:bl:conc}
%
\psi(k)=\biggl\|
\frac1{\lmes \set'\cap\{\lst\element k\}\rmes }
\mathop{\sum\ \e_\element}\limits_{\set'\cap\{\lst\element k\}}-
\frac 1{\sigma_k}\sum_{j=1}^k\delta_{\element_j}\e_{\element_j}
\biggr\|_\infty\tol_{k\to\infty}0.
\end{equation}
%
\end{lem}
%
\dem
Note that 
$$
\mathop{\sum\ \e_\element}\limits_{\set'\cap\{\lst\element k\}}= 
\sum_{j=1}^k\xi_{\element_j}\e_{\element_j}\quad,\quad
\mes{\set'\cap\{\lst\element k\}}
=\sum_{j=1}^k\xi_{\element_j}.
$$ 
Center the $\xi_\element$ by putting 
$f=\sum_{j=1}^k(\xi_{\element_j}-\delta_{\element_j})\e_{\element_j}$. Then
%
\begin{eqnarray*}
%
\psi(k)&\le&\Bigl\|
\Bigl(\mes{\set'\cap\{\lst\element k\}}^{-1}-\sigma_k^{-1}\Bigr)
\sum_{j=1}^k\xi_{\element_j}\e_{\element_j}\Bigr\|_\infty+
\|\sigma_k^{-1}f\|_\infty\\
&\le&
\sigma_k^{-1}\,\biggl|\frac {\delta_{\element_1}+\dots+\delta_{\element_k}}
{\xi_{\element_1}+\dots+\xi_{\element_k}} - 1\biggr|\ 
\sum_{j=1}^k\xi_{\element_j}+
\sigma_k^{-1}\|f\|_\infty\le2\sigma_k^{-1}\|f\|_\infty.
\end{eqnarray*}
%
Let $R=\{t\in\T:\,t^{4|\element_k|}=1\}$ and $u\in\T$ such that
$|f(u)|=\|f\|_\infty$. Let $t\in R$ be at
minimal distance of $u$: then $|t-u|\le\pi/4|\element_k|$. By
Bernstein's theorem,
$$
\|f\|_\infty-|f(t)| = |f(u)-f(t)| \le |t-u| \, \|f'\|_\infty \le
\frac45 \|f\|_\infty
$$
$$
\|f\|_\infty\le5\sup_{t\in R}|f(t)|.
$$
(For an optimal bound, cf.\ \cite[\S I.4, Lemma 8]{me72}.) 
%As $\mes{R}=4|\element_k|$ and 
For each $t\in R$, the random variables
$X_j=(\xi_{\element_j}-\delta_{\element_j})\e_{\element_j}(t)$ satisfy
$(\ref{ergo:bern:hyp})$, so that
$$
\P{|f(t)| \ge a}<4\exp(-a^2/4(\sigma_k+a)).
$$ 
As $\mes{R}=4|\element_k|$, 
$$
\P{\|f\|_\infty \ge 5a}\le\P{\sup_{t\in
  R}|f(t)| \ge a}<4|\element_k|\cdot4\exp(-a^2/4(\sigma_k+a)).
$$
Put $a_k=(12\sigma_k\log |\element_k|)^{1/2}$. Then
$a_k\ll\sigma_k$: therefore
%: let $N$ such that $a_k\le\sigma_k$ pour $k\ge N$. Donc
$$
\P{\|f\|_\infty \ge 5a_k} \preccurlyeq |\element_k|^{-2}
$$
and by the Borel--Cantelli lemma, 
$$
\sigma_k^{-1}\|f\|_\infty\preccurlyeq a_k/\sigma_k\tol0\hbox{ almost surely}.
\eqno{\ecks}
$$

\rem
The hypothesis in Lemma \ref{ergo:bourgain-lem} contains implicitly a
restriction on the lacunarity of $\set$. If $\sigma_k \gg \log
|\element_k|$, then necessarily $\log|\element_k| \ll k$ and $\set[t]
\gg \log t$. In particular, $\set$ cannot be a Sidon set by
\cite[Cor.\ of Th.\ 3.6]{ru60}. 

\vskipa
We now state and prove the equidistributed counterpart of Theorem
\ref{lambda:katz-gen}.
%
\begin{thm}\label{ergo:bourgain-gen}
%
Let \DLL/ be equidistributed\index{equidistributed set} 
\(\vs weakly\index{weakly equidistributed set}\), and ordered by
increasing absolute value. Do Construction \ref{def:constr} and
suppose that $\delta_{\element_j}$ decreases with $j$. Put
$\sigma_k=\delta_{\element_1}+\dots+\delta_{\element_k}$. If $\sigma_k
\gg \log|\element_k|$, then $\set'$ is almost surely equidistributed
\(\vs weakly\). Note that this is in particular the case if

\bm{a}
$\ds\delta_{\element_k} \gg (|\element_k|-|\element_{k-1}|)/
{|\element_{k-1}|}$; 

\bm{b}
$\set$ has polynomial growth and $\delta_{\element_k}\gg k^{-1}$.
%
\end{thm}
%
\dem
Lemma \ref{ergo:bourgain-lem} shows that almost surely
\Ref{ergo:bl:conc} holds. It remains to show that 
$$
\lim \frac 1 {\sigma_k}\sum_{j=1}^k\delta_{\element_j}\e_{\element_j} = \lim
\frac 1{k} \sum_{j=1}^k\e_{\element_j}, 
$$
\ie that the matrix summing method\index{matrix summing method} 
$(a_{k,j})$ given by 
$$
a_{k,j}=\left\{
\begin{array}{ll}
\delta_{\element_j}/\sigma_k&\mbox{ if }j\le k\\
0&\mbox{ otherwise}
\end{array}\right.
$$
is regular and stronger than the Ces\`aro 
method\index{Ces\`aro summing method} $C_1$ by arithmetic
means. This is the case because $a_{k,j}\ge0$,
$\sum_ja_{k,j}=1$ and (cf.\ \cite[\S52, Th.\ I]{ze58}) 
$$
\forall k\quad\sum_j j|a_{k,j}-a_{k,j+1}|=
\sum_j j(a_{k,j}-a_{k,j+1})=1<\infty 
$$
since $(a_{k,j})$ decreases with $j$ for each $k$.

$(a)$ In this case $\delta_{\element_k} \gg
\log|\element_k|-\log|\element_{k-1}|$ and thus $\sigma_k \gg
\log|\element_k|$. 

$(b)$
In this case, $\sigma_k \gg \log k \succcurlyeq \log
|\element_k|$.
\eck\vskipb 

In conclusion, we obtain, by combining Theorems \ref{lambda:katz-gen} and
\ref{ergo:bourgain-gen}, the principal result of this chapter.
%
\begin{cor}\label{ergo:COR}
%
Let \DL/ be \ud/ \(\vs weakly\) and do Construction
\ref{lambda:constr}. Then
$\set'$ is almost surely \EL{p}\index{Lambda(p) set@\EL{p} set!for all $p$}  for all
$p$ and at the same time \ud/\index{equidistributed set}
\(\vs weakly\index{weakly equidistributed set}\) in the two following cases:

\bm{i}
$\set$ is a set of regular polynomial growth, $\{I_j\}$ is the dyadic
Littlewood--Paley partition \Ref{poly:lpd} and one may choose
$\{\ell_j\}$ such that $1 \ll \log\ell_j \ll j$ and
$\ell_j/\mes{\set_j}$ decreases eventually. 

\bm{ii}
$\set$ is a set of polynomial growth, $\{I_j\}$ is the gross
Littlewood--Paley partition \Ref{poly:lpg} and one may choose
$\{\ell_j\}$ such that $\ell_j/\mes{\set_j}$ decreases eventually
and $\ell_j 
\gg \log p_{j+1}$ while 
$\log\ell_j \ll \log p_j $. This is the case if we put 
$p_j=2^{j!}$ and $\ell_j=\min((j+2)!,\mes{\set_j})$.
%
\end{cor}
%
\dem
In each case $\log\ell_j \ll \log\lmes
I_j\rmes$. Let us show that the hypothesis of Theorem
\ref{ergo:bourgain-gen} is verified. If
$\element_k\in\set_j\subseteq I_j$, then $|\element_k|\le p_j $ and 
$$
\sigma_k\ge\sum_{i=1}^{j-1}\sum_{\element\in\set_i}\delta_\element=
\sm\ell{j-1}
$$
and in each case $\ell_{j-1} \gg \log p_j -\log p_{j-1}$.

Let us make sure in $(ii)$ that our choice for $p_j$ and $\ell_j$
is accurate. Indeed, there is an $\eps>0$ such that
$\mes{\set_j}\succcurlyeq2^{\eps j!}$. Thus $(j+2)!\ll\mes{\set_j}$ and
$\ell_j=(j+2)!$ for large $j$. Note further that $(j+2)!\gg(j+1)!$ while
$\log(j+2)!\ll j!$. Finally
$$
\frac {\ell_{j+1}} {\mes{\set_{j+1}}} \preccurlyeq \frac {(j+3)!}
{2^{\eps(j+1)!}}  \preccurlyeq \frac {j\ell_j}{2^{\eps(j+1)!}}
\ll \frac {\ell_j} {2^{j!}} \preccurlyeq \frac {\ell_j} {\mes{\set_j}},
$$
so that $\ell_j/\mes{\set_j}$ decreases eventually.\eck
%
%
%
%
%
%
%
%
%\baselineskip=12pt
%\bibliography{references}
%
%
%
%
%
%
%\end{document}


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\end{thebibliography}
}

\baselineskip=13pt minus 1pt
\chapter*{Index of notation}
\markboth{index of notation}{index of notation}
\addcontentsline{toc}{chapter}{Index of notation}

%\begin{tabular}{rl}
\halign{\hfil#&\quad#\hfil\cr
%symbols
$\mes{B}$&cardinal of $B$\cr
$X_E$&space of $X$-functions with spectrum in $E$\cr
$\widehat{f}$&Fourier transform of $f$: $\widehat{f}(n)=\int
 f(t)\e_{-n}(t)dm(t)$\cr 
$x\choose\alpha$&multinomial number, \S\ref{ss:almost}\cr
$\XE$&pairing of the arithmetical relation $\zeta$ against
 the spectrum $E$, \S\ref{ss:mubs:gen}\cr
$u_n\preccurlyeq v_n$&$|u_n|$ is bounded by $C|v_n|$ for some $C$\cr
$u_n\ll v_n$&$u_n$ is negligible with respect to $v_n$\cr
$E[t]$&distribution function of $E\se\Z$: $E[t]=\mes{E\cap[-t,t]}$\cr
$(x_{rc})$&the operator $x$ on $\ell_2$ viewed as matrix\cr
%$\vcenter{\hbox{\hfil$q_1\to q_2$}\par
%\hbox{\hfil$q_1\downto q_2$}}\rlap{$\Bigr\}$}$&
$q_1\to q_2$&segment from $q_1$ to $q_2$, where
$q_1,q_2\in\N\times\N$\cr 
$q_1\downto q_2$&segment from $q_1$ to $q_2$, where
$q_1,q_2\in\N\times\N$\cr 
\noalign{\vskip6.5pt minus 0.5pt}
%ciphers
$1$-\ubs/&$1$-unconditional basic sequence of characters, Def.\
\ref{mub:def}$(i)$\cr \noalign{\vskip6.5pt minus 0.5pt}
%a
$A(\T)$&disc algebra $\SC{\sN}$\cr
$\Alpha_n,\Alpha_n^m$&sets of multi-indices viewed as arithmetic
 relations, \S\ref{ss:isom}\cr
a.s.&approximating sequence, Def.\ \ref{block:def}\cr\noalign{\vskip6.5pt minus 0.5pt}
%b
$B_X$&unit ball of the Banach space $X$\cr\noalign{\vskip6.5pt minus 0.5pt}
%c
$\SC{}$&space of continuous functions on $\T$\cr
$c_0(\T)$&space $\{f\colon\T\to\T:\forall\eps>0\ \exists A\se\T$ finite
$|f|<\eps$ outside $A\}$\cr
c.b.&completely bounded\cr\noalign{\vskip6.5pt minus 0.5pt}
%d
$\D$&set of real signs $\{-1,1\}$\cr
$\Delta T_k$&difference sequence of the $T_k$: $\Delta
 T_k=T_k-T_{k-1}$ ($T_0=0$)\cr\noalign{\vskip6.5pt minus 0.5pt}
%e
$\E X$&expectation of the random variable $X$\cr 
$\e_n$&character of $\T$: $\e_n(z)=z^n$ for $z\in\T$, $n\in\Z$\cr
$\e_{rc}$&matrix entry seen as operator on $\ell_2$\cr\noalign{\vskip6.5pt minus 0.5pt}
%f
\fdd/&finite dimensional decomposition, Def.\ \ref{block:def}\cr\noalign{\vskip6.5pt minus 0.5pt}
%h
$H^1(\T)$&Hardy space $\SL{1}{\sN}$\cr\noalign{\vskip6.5pt minus 0.5pt}
%i
\I{n}&arithmetical property of almost independence, Def.\
 \ref{mub:def:ar}\cr
$\Id$&identity\cr
i.i.d.&independent identically distributed, \S\ref{ss:proba:two}\cr\noalign{\vskip6.5pt minus 0.5pt}
%j
\J{n}&arithmetical property of block independence, Def.\
 \ref{arith:def}\cr \noalign{\vskip6.5pt minus 0.5pt}
%l
$\script{L}(X)$&space of bounded linear operators on the Banach space $X$\cr
$\SLP{}$&Lebesgue space of $p$-integrable functions on $\T$\cr
\lpap/&$p$-additive approximation property, Def.\ \ref{str:def}\cr
\lpmap/&metric $p$-additive approximation property, Def.\
 \ref{str:def}\cr 
$\Lambda(p)$&Rudin's class of lacunary sets, Def.\ \ref{mub:sido}\cr\noalign{\vskip6.5pt minus 0.5pt}
%m
$\script{M}_p$&functional property of Fourier block $p$-additivity,
Lemma \ref{sbd:lpmap:lem}$(ii)$\cr
$\script{M}(\T)$&space of Radon measures on $\T$\cr
$m[A]$&measure of $A\se\T$\cr
$(m_p(\tau))$&functional property of $\tau$-$p$-additivity,
Def.\ \ref{block:strong:dfn}$(i)$\cr
$(m_p(T_k))$&functional property of commuting block $p$-additivity,
Def.\ \ref{block:strong:dfn}$(ii)$\cr\noalign{\vskip6.5pt minus 0.5pt}
%o
$\osc f$&oscillation of $f$\cr\noalign{\vskip6.5pt minus 0.5pt}
%p
$\P{A}$&probability of the event $A$\cr 
$\PT{}$&space of trigonometric polynomials on $\T$\cr
$\pi_j$&projection of $X_E$, $E=\{n_k\}$, onto $X_{\{\lst n j\}}$\cr
$\pi_F$&projection of $X_E$ onto $X_F$\cr\noalign{\vskip6.5pt minus 0.5pt}
%s
$\U$&real ($\U=\D$) or complex ($\U=\T$) choice of signs\cr
$S^p$&Schatten class of operators\cr
$S^p_I$&entry subspace of $S^p$\cr\noalign{\vskip6.5pt minus 0.5pt}
%t
$(\T,dm)$&unit circle in $\C$ with its normalized Haar measure\cr
$\tau_f$&topology of pointwise convergence of the Fourier 
coefficients, Lemma\ \ref{blockapp:lem}$(i)$\cr\noalign{\vskip6.5pt minus 0.5pt}
%u
\UP/&functional property of Fourier block unconditionality, Def.\
 \ref{block:block:def}\cr
$(u(\tau))$&functional property of $\tau$-unconditionality, Def.\
\ref{block:def:u}$(i)$\cr
$(u(T_k))$&functional property of commuting block 
unconditionality, Def.\ \ref{block:def:u}$(ii)$\cr
\uap/&unconditional approximation property, Def.\
 \ref{block:def}\cr 
\ubs/&unconditional basic sequence, Def.\ \ref{mub:def}\cr 
\umap/&metric unconditional approximation property, Def.\
 \ref{block:def}\cr
\umbs/&metric unconditional basic sequence, Def.\
 \ref{mub:def}\cr \noalign{\vskip6.5pt minus 0.5pt}
%z
$\Zeta^m,\Zeta_n^m$&sets of multi-indices viewed as arithmetic relations,
 \S\ref{ss:isom}\cr
%$E=\{n_k\}$&spectrum viewed as a subset of $\Z$\cr
%$\script{R}_q$&partition of the considered set of arithmetic
%relations, \S\ref{ss:isom},\ref{ss:almost},\ref{ss:block}
%
}
%\end{tabular}
\vfill\pagebreak

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% \advance\oddsidemargin by -20mm
% \advance\evensidemargin by -20mm

% \baselineskip=13pt minus 1.2pt
\addcontentsline{toc}{chapter}{Index}
\begin{flushleft}
  \input these.ind
\end{flushleft}

\end{document}

