Thèse de Doctorat de l’Université Paris VI Mathématiques

Stefan Neuwirth

Multiplicateurs et analyse fonctionnelle

Thèse soutenue le 19 janvier 1999 devant le jury composé de

M^{me} Myriam Déchamps,
M. Jean Esterle,
M. Gilles Godefroy,
M. Daniel Li,
M^{me} Françoise Piquard,
M. Hervé Queffélec et
M. Jean Saint Raymond

Cette thèse a été écrite sous la direction de M. Gilles Godefroy et de M. Daniel Li. Elle est redevable de leur culture, de la précision de leur écoute, ainsi que de leurs multiples questions.

M^{me} Françoise Piquard et M. Jean Esterle ont accepté de lire cette thèse et de rédiger un rapport. M^{me} Myriam Déchamps, M. Hervé Queffélec et M. Jean Saint Raymond ont bien voulu faire partie du jury. Leur présence m’honore.

Ce travail scientifique a été l’occasion de nombreuses rencontres, d’abord à Paris, puis à Columbia dans le Missouri, enfin à Besançon. Je les garderai dans mon cœur.

Table des matières

Résumé de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Zusammenfassung der Dissertation auf deutsch . . . . . . . . . . . . . . . 8 Abstract of the thesis in English . . . . . . . . . . . . . . . . . . . . . . . 9

I Introduction 11

1 Position du problème . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1 Chapitre II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Chapitre III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Chapitre IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Inconditionnalité métrique en analyse de Fourier . . . . . . . . . . . 15

2.1 Propriété d’approximation inconditionnelle métrique . . . . . 15 2.1.1 Amorce et queue d’un espace de Banach . . . . . . 15 2.1.2 Amorce et queue en termes de spectre de Fourier . . 16

2.2 Norme de multiplicateurs et conditions combinatoires …. 17 2.2.1 Suites basiques 1-inconditionnelles complexes …. 18 2.2.2 Suites basiques 1-inconditionnelles réelles . . . . . . 18 2.2.3 Suites basiques inconditionnelles métriques . . . . . 19 2.2.4 Propriété d’approximation inconditionnelle métrique 19 2.2.5 Deux exemples . . . . . . . . . . . . . . . . . . . . . 20

2.3 Impact de la croissance du spectre . . . . . . . . . . . . . . . 20 3 Suites basiques 1-inconditionnelles d’entrées de matrice . . . . . . . 21 4 Constructions aléatoires à l’intérieur de suites lacunaires . . . . . . . 22

II Metric unconditionality and Fourier analysis 23 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Metric unconditional basic sequences of characters (umbs) . . . . . . 27

2.1 Definitions. Isomorphic case . . . . . . . . . . . . . . . . . . . 27 2.2 Isometric case: 1-unconditional basic sequences of characters 29 2.3 Almost isometric case. A computation . . . . . . . . . . . . . 32 2.4 Almost independent sets of integers. Main theorem . . . . . . 34

3 Examples of metric unconditional basic sequences . . . . . . . . . . . 35 3.1 General considerations . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Geometric sequences . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Algebraic and transcendental numbers . . . . . . . . . . . . . 36 3.4 Polynomial sequences . . . . . . . . . . . . . . . . . . . . . . 37

4 Metric unconditional approximation property (umap) . . . . . . . . . 37 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Characterization of (umap). Block unconditionality . . . . . . 39 4.3 Main theorem: convex combinations of multipliers . . . . . . 40

5 The p-additive approximation property ℓ_{p}-(ap) . . . . . . . . . . . . 42 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Some consequences of ℓ_{p}-(ap) . . . . . . . . . . . . . . . . . . 44 5.3 Characterization of ℓ_{p}-(map) . . . . . . . . . . . . . . . . . . . 44

5.4 Subspaces of L^{p} with ℓ_{p}-(map) . . . . . . . . . . . . . . . . . 45 6 (uap) and (umap) in translation invariant spaces . . . . . . . . . . . . 47

6.1 General properties. Isomorphic case . . . . . . . . . . . . . . 47 6.2 Characterization of (umap) and ℓ_{p}-(map) . . . . . . . . . . . . 48

7 Property (umap) and arithmetical block independence . . . . . . . . 51 7.1 Property of block independence . . . . . . . . . . . . . . . . . 51 7.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8 Examples for (umap): block independent sets of characters . . . . . . 55 8.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . 55 8.2 Geometric sequences . . . . . . . . . . . . . . . . . . . . . . . 56 8.3 Symmetric sets . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.4 Algebraic and transcendental numbers . . . . . . . . . . . . . 57 8.5 Polynomial sequences . . . . . . . . . . . . . . . . . . . . . . 57

9 Positive results: parity and a sufficient growth condition . . . . . . . 58 9.1 𝒞_{{3^{k}}}(𝕋) has real (umap) because 3 is odd . . . . . . . . . . . 58 9.2 Growth conditions: the case L^{p}(𝕋), p an even integer . . . . . 59 9.3 A general growth condition . . . . . . . . . . . . . . . . . . . 59

10 An excursion: estimation of the Sidon constant . . . . . . . . . . . . 60 10.1 Sidon constant of sets with three elements . . . . . . . . . . . 60 10.2 Sidon constant of Hadamard sets . . . . . . . . . . . . . . . . 64

11 Density conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 12 Unconditionality vs. probabilistic independence . . . . . . . . . . . . 66

12.1 Cantor group . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 12.2 Two notions of approximate probabilistic independence … 66

13 Summary of results. Remarks and questions . . . . . . . . . . . . . . 68 13.1 The case X = L^{p}(𝕋) with p an even integer . . . . . . . . . . 68 13.2 Cases X = L^{p}(𝕋) with p not an even integer and X = 𝒞(𝕋) . 69 13.3 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

IIIUnconditional entry basic sequences 71 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2 Rectangular polygons in the two-dimensional integer lattice . . . . . 72 3 1-unconditional basic sequences of entries . . . . . . . . . . . . . . . 75 4 Two remarks on a paper by Harcharras . . . . . . . . . . . . . . . . 77

4.1 The c.b. unconditionality constant of sum sets . . . . . . . . 77 4.2 Two new sufficient conditions for Λ(2s) sets . . . . . . . . . . 78

IV Random constructions inside lacunary sets 79 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2 Equidistributed and Λ(p) sets . . . . . . . . . . . . . . . . . . . . . . 81 3 Sets with polynomial growth . . . . . . . . . . . . . . . . . . . . . . 82 4 Sets that are Λ(p) for all p . . . . . . . . . . . . . . . . . . . . . . . . 83 5 Equidistributed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Bibliography 88

Index of notation 95

Index 97

Résumé de la thèse

Nous étudions plusieurs propriétés fonctionnelles d’inconditionnalité en les exprimant à l’aide de multiplicateurs. La première partie est consacrée à l’étude de phénomènes d’inconditionnalité isométrique et presqu’isométrique dans les espaces de Banach séparables. Parmi ceuxci, la notion la plus générale est celle de “propriété d’approximation inconditionnelle métrique”. Nous la caractérisons parmi les espaces de Banach de cotype fini par une propriété simple d’“inconditionnalité par blocs”. En nous ramenant à des multiplicateurs de Fourier, nous étudions cette propriété dans les sous-espaces des espaces de Banach de fonctions sur le cercle qui sont engendrés par une suite de caractères e^{int}. Nous étudions aussi les suites basiques inconditionnelles isométriques et presqu’isométriques de caractères, en particulier les ensembles de Sidon de constante asymptotiquement 1. Nous obtenons dans chaque cas des propriétés combinatoires sur la suite. La propriété suivante des normes L^{p} est cruciale pour notre étude: si p est un entier pair, ∫ |f|^{p} = ∫ |f^{p/2}|^{2} = ∑ |f̂^{p/2}(n)|^{2} est une expression polynômiale en les coefficients de Fourier de f et f̄. Nous proposons d’ailleurs une estimation précise de la constante de Sidon des ensembles à la Hadamard. La deuxième partie étudie les multiplicateurs de Schur: nous caractérisons les suites basiques inconditionnelles isométriques d’entrées de matrice e_{ij} dans la classe de Schatten S^{p}. Les propriétés combinatoires que nous obtenons portent sur les chemins dans le réseau ℕ × ℕ à sommets dans cet ensemble. La troisième partie étudie le rapport entre la croissance d’une suite d’entiers et les propriétés harmoniques et fonctionnelles de la suite de caractères associée. Nous montrons en particulier que toute suite polynômiale, ainsi que la suite des nombres premiers, contient un ensemble Λ(p) pour tout p qui n’est pas de Rosenthal.

Zusammenfassung der Dissertation

Verschiedene funktionalanalytische Unbedingtheitseigenschaften werden mittels Multiplikatoren untersucht. Teil I beschreibt die Begriffe isometrischer und fast isometrischer Unbedingtheit in separablen Banachräumen. Am allgemeinsten ist die metrische unbedingte Approximationseigenschaft gefasst. Wir charakterisieren diese für Banachräume mit endlichem Kotyp durch eine einfache “blockweise” Unbedingtheit. Daraufhin betrachten wir genauer den Fall von Funktionenräumen auf dem Einheitskreis, die durch eine Folge von Frequenzen e^{int} aufgespannt werden. Wir untersuchen isometrisch und fast isometrisch unbedingte Basisfolgen von Frequenzen, unter anderem Sidonmengen mit einer Konstante asymptotisch zu 1. Für jeden Fall erhalten wir kombinatorische Eigenschaften der Folge. Die folgende Eigenschaft der L^{p} Normen ist entscheidend für diese Arbeit: Ist p eine gerade Zahl, so ist ∫ |f|^{p} = ∫ |f^{p/2}|^{2} = ∑ |f̂^{p/2}(n)|^{2} ein polynomialer Ausdruck der Fourierkoeffizienten von f und f̄. Des weiteren erhalten wir eine genaue Abschätzung 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 × 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 zugehörigen Folge von Frequenzen. Wir zeigen insbesondere, dass jede polynomiale Folge, sowie die Primzahlenfolge, eine Unterfolge enthält, die zwar Λ(p) für jedes p aber keine Rosenthalmenge ist.

Abstract of the thesis

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^{int}. 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 L^{p} norm for even p: ∫ |f|^{p} = ∫ |f^{p/2}|^{2} = ∑ |f̂^{p/2}(n)|^{2} is a polynomial expression in the Fourier coefficients of f and f̄. As a byproduct, we get a sharp estimate of the Sidon constant of sets à 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 × 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 Λ(p) for all p but is not a Rosenthal set. This holds also for the sequence of primes.

Chapitre I

Introduction

1 Position du problème

Cette thèse se situe au croisement de l’analyse fonctionnelle et de l’analyse harmonique. Nous allons donner des éléments de réponse à la question générale suivante. Question 1.1 Quelle est la validité de la représentation

f ∼ ∑ ϱ_{q} e^{iϑ_{q}} e_{q}  (1)

de la fonction f comme série de fréquences e_{q} d’intensité ϱ_{q} et de phase ϑ_{q} ? Les réponses seront donnés en termes de l’espace de fonctions X ∋ f et du spectre E ⊇ {q : ϱ_{q} > 0}.

1.1 Chapitre II

Considérons par exemple les deux questions classiques suivantes dans le cadre des espaces de Banach homogènes de fonctions sur le tore 𝕋, des fréquences de Fourier e_{q}(t) = e^{iqt} et des coefficients de Fourier

ϱ_{q} e^{iϑ_{q}} = ∫ e_{−q} f = f̂(q).

Question 1.1.1 Est-ce que pour les fonctions f ∈ X à spectre dans E

∥f − ∑_{|q|≤n} ϱ_{q} e^{iϑ_{q}} e_{q}∥_{X} ⟶_{n→∞} 0 ?

Cela revient à demander: est-ce que la suite {e_{q}}_{q∈E} rangée 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} → X_{E} définie par

T_{n} e_{q} = \{ \begin{matrix}e_{q} & si |q| ≤ n \\ 0 & sinon \end{matrix}

est-elle uniformément bornée sur n ? Soit E = ℤ. Un élément de réponse classique est le suivant.

∥T_{n}∥_{L^{2}(𝕋)→L^{2}(𝕋)} = 1 , ∥T_{n}∥_{L^{1}(𝕋)→L^{1}(𝕋)} = ∥T_{n}∥_{𝒞(𝕋)→𝒞(𝕋)} ≍ log n.

On sait de plus que les T_{n} sont aussi uniformément bornés sur L^{p}(𝕋), 1 < p < ∞. Question 1.1.2 Est-ce que la somme de la série ∑ ϱ_{q} e^{iϑ_{q}} e_{q} dépend de l’ordre dans lequel on somme les fréquences ? Cette question est équivalente à la suivante: la nature de ∑ ϱ_{q} e^{iϑ_{q}} e_{q} dépend-elle des phases ϑ_{q} ? En termes fonctionnels, {e_{q}}_{q∈E} forme-t-elle une suite basique inconditionnelle dans X ? Cette question s’énonce aussi en termes de multiplicateurs relatifs: la famille des T_{ϵ} : X_{E} → X_{E} avec

T_{ϵ} e_{q} = ϵ_{q} e_{q} et ϵ_{q} = ±1

est-elle uniformément bornée sur les choix de signes ϵ ? Un élément de réponse classique est le suivant. Soit E = ℤ. Alors

∥T_{ϵ}∥_{L^{2}(𝕋)→L^{2}(𝕋)} = 1;

si p ≠ 2, il existe un choix de signes ϵ tel que T_{ϵ} n’est pas borné sur L^{p}(𝕋). Question 1.1.3 Peut-on améliorer ce phénomène en restreignant le spectre E ? Cette question mène à l’étude des sous-ensembles lacunaires de ℤ, et a été traitée en détail par Walter Rudin. Nous choisissons la notion de multiplicateur relatif comme dictionnaire entre l’analyse harmonique et l’analyse fonctionnelle. Nous développons une technique pour le calcul de la norme de familles {T_{ϵ}} de multiplicateurs relatifs. Celle-ci nous permet de traiter les questions suivantes. Question 1.1.4 Est-ce que la norme de f ∈ X_{E} dépend seulement de l’intensité ϱ_{q} de ses fréquences e_{q}, et non pas de leur phase ϑ_{q} ? Cela revient à demander si {e_{q}}_{q∈E} est une suite basique 1-inconditionnelle complexe dans X. Question 1.1.5 Est-ce que l’on a pour tout choix de signes “réel” ±

∥∑_{q∈E} ±a_{q} e_{q}∥_{X} = ∥∑_{q∈E} a_{q} e_{q}∥_{X} ?

En d’autres mots, est-ce que {e_{q}}_{q∈E} est une suite basique 1-inconditionnelle réelle dans X ? La réponse est décevante dans le cas des espaces L^{p}(𝕋), p non entier pair: seules les fonctions dont le spectre a au plus deux éléments vérifient ces deux propriétés. Pour mieux cerner le phénomène, nous proposons d’introduire la question presqu’isométrique suivante. Question 1.1.6 Est-ce que la norme de f ∈ X_{E} dépend arbitrairement peu de la phase ϑ_{q} de ses fréquences e_{q} ? De manière précise, dans quel cas existe-t-il, pour chaque ε > 0, un sous-ensemble F ⊆ E fini tel que

∥ ∑_{q∈E\F} ϱ_{q} e^{iϑ_{q}} e_{q}∥_{X} ≤ (1 + ε)∥ ∑_{q∈E\F} ϱ_{q} e_{q}∥_{X} ?

Dans le cas X = 𝒞(𝕋), cela signifiera que E est un ensemble de constante de Sidon “asymptotiquement 1”. De même, peut-on choisir pour chaque ε > 0 un ensemble fini F tel que pour tout choix de signe “réel” ±

∥ ∑_{q∈E\F} ±a_{q} e_{q}∥_{X} ≤ (1 + ε)∥ ∑_{q∈E\F} a_{q} e_{q}∥_{X} ?

Toutes ces questions s’agrègent autour d’un fait bien connu: sommer la série de Fourier de f est une très mauvaise manière d’approcher la fonction f dès que l’erreur considérée n’est pas quadratique. On sait qu’il est alors utile de rechercher des méthodes de sommation plus lisses, c’est-à-dire d’autres suites approximantes plus régulières. Il s’agit là de suites d’opérateurs de rang fini sur X_{E} qui approchent ponctuellement l’identité de X_{E}. Nous pourrons toujours supposer que ces opérateurs sont des multiplicateurs. Une première question est la suivante. Question 1.1.7 Existe-t-il une suite approximante {T_{n}} de multiplicateurs idempotents ? Cela revient à demander: existe-t-il une décomposition de X_{E} en sousespaces X_{E_{k}} de dimension finie avec

X_{E} = ⨁ X_{E_{k}} et A_{k} : X_{E} → X_{E_{k}} , e_{q} ↦ \{ \begin{matrix}e_{q} & si q ∈ E_{k} \\ 0 & sinon \end{matrix}  (2)

telle que la suite des T_{n} = A_{1} + … + A_{n} est uniformément bornée sur n ? Soit E = ℤ. Alors la réponse est identique à la réponse de la question 1.1.1. Mais nous pouvons produire dans ce cadre plus général des décompositions inconditionnelles de X_{E} en réponse à la question suivante. Question 1.1.8 Pour quels espaces X et spectres E existe-t-il une décomposition comme ci-dessus telle que la famille des multiplicateurs

∑_{k=1}^{n} ϵ_{k}A_{k} avec n ≥ 1 et ϵ_{k} = ±1  (3)

est uniformément bornée ? Littlewood et Paley ont montré que la partition de ℤ en ℤ = ⋃ E_{k} avec E_{0} = {0} et E_{k} = {j : 2^{k−1} ≤ |j| < 2^{k}} donne une décomposition inconditionnelle des espaces L^{p}(𝕋) avec 1 < p < ∞. D’après la réponse à la question 1.1.7, ce n’est pas le cas a fortiori des espaces L^{1}(𝕋) et 𝒞(𝕋). Une étude fine de telles partitions a été entreprise par Kathryn Hare et Ivo Klemes. Notre technique permet de traiter la question suivante. Question 1.1.9 Pour quels espaces X et spectres E existe-t-il une décomposition du type (2) telle que

∥∑ ϵ_{k}A_{k}f∥_{X} = ∥f∥_{X} pour tout choix de signes ϵ_{k} ?

La réponse dépendra de la nature du choix de signes, qui peut être réel ou complexe. Il est instructif de noter que l’espace de Hardy H^{1}(𝕋) n’admet pas de décomposition du type (2). H^{1}(𝕋) admet néanmoins des suites approximantes de multiplicateurs et il existe même des suites approximantes de multiplicateurs inconditionnelles au sens où la famille (3) est uniformémemt bornée. Cela motive la question suivante, qui est la plus générale dans notre contexte. Question 1.1.10 Quels sont les espaces X et spectres E tels que pour chaque ε > 0 il existe une suite approximante {T_{n}} sur X_{E} telle que

sup_{signes ϵ_{n}}∥∑ ϵ_{n}(T_{n} − T_{n−1})∥_{X} ≤ 1 + ε

En termes fonctionnels, X_{E} a-t-il la propriété d’approximation inconditionnelle métrique ? Il faudra distinguer le cas des signes complexes et réels.

1.2 Chapitre III

Nous montrons que notre technique de calcul s’applique mutatis mutandis aux multiplicateurs de Schur. La représentation (1) est alors la représentation matricielle: on note e_{q} = e_{rc} l’entrée de matrice en q = (r, c) ∈ ℕ × ℕ, c’est-à-dire l’opérateur sur ℓ_{2} qui envoie son cième vecteur de base sur son rième vecteur de base et de matrice (δ_{k}^{r}δ_{l}^{c})_{k,l≥0}. On considère donc la validité de

x ∼ ∑ ϱ_{q} e^{iϑ_{q}} e_{q} avec les coefficients de matrice x_{q} = ϱ_{q} e^{iϑ_{q}} = tr e_{q}^{∗} x

pour x opérateur sur ℓ_{2}. Soit p ≥ 1 et I ⊆ ℕ × ℕ. Notre étude se fera en termes de la classe de Schatten S^{p} de x et du support I de la matrice (x_{q}) associée à x. Nous dirons que x est à entrées dans I si I ⊇ {q : x_{q} ≠ 0}. Question 1.2.1 Est-ce que la norme de x ∈ S^{p} à entrées dans I dépend seulement du module ϱ_{q} de ses coefficients de matrice et non pas de leur argument ϑ_{q} ? Cela revient à demander: quelles sont les suites basiques d’entrées 1-inconditionnelles dans S^{p} ? En termes de multiplicateurs de Schur, la question se pose ainsi. Pour quels ensembles d’entrées I les

T_{ϵ} : S_{I}^{p} → S_{I}^{p} , e_{q} ↦ ϵ_{q} e_{q} avec |ϵ_{q}| = 1

sont-ils tous des isométries ?

1.3 Chapitre IV

Nous étudions le rapport entre la croissance d’une suite {n_{k}} = E ⊆ ℤ et deux de ses propriétés harmoniques et fonctionnelles éventuelles, i. e. toute fonction intégrable à spectre dans E est en fait p-intégrable pour tout p < ∞: E est un ensemble Λ(p) pour tout p; toute fonction mesurable bornée à spectre dans E est en fait continue à un ensemble de mesure nulle près: E est un ensemble de Rosenthal. Nous sommes en mesure de dresser le tableau suivant selon la croissance polynômiale: n_{k} ≼ k^{d} pour un d < ∞, surpolynômiale: n_{k} ≫ k^{d} pour tout d ≥ 1, sous-exponentielle: log n_{k} ≪ k, géométrique: lim inf |n_{k+1}/n_{k}| > 1.

croissance polynômiale surpolynômiale&sous-exponentielle géométrique
E Λ(p) ∀p  non  presque toujours  oui
E Rosenthal  presque jamais  oui

Tableau 1.3.1

Li montre qu’effectivement il existe un ensemble Λ(p) pour tout p qui n’est pas de Rosenthal. Nous traitons les deux questions suivantes. Question 1.3.2 Le schéma ci-dessus reste-t-il valable si on considère à la place de l’ensemble des sous-ensembles E de ℤ l’ensemble des sous-ensembles E d’une suite à croissance polynômiale ? Question 1.3.3 Si E n’est pas un ensemble de Rosenthal, E contient-il un ensemble à la fois Λ(p) pour tout p et non Rosenthal ?

2 Inconditionnalité métrique en analyse de Fourier

Nous répondons dans ce chapitre aux questions 1.1.4, 1.1.5, 1.1.6, 1.1.9 et 1.1.10. Comme ces questions distinguent les choix de signe réel et complexe, nous proposons pour la fluidité de l’exposé de fixer un choix de signes 𝕊 qui sera 𝕊 = 𝕋 = {ϵ ∈ ℂ : |ϵ| = 1} dans le cas complexe et 𝕊 = 𝔻 = {−1, 1} dans le cas réel.

2.1 Propriété d’approximation inconditionnelle métrique

Seule la question 1.1.10 n’impose pas au préalable de forme particulière à la suite de multiplicateurs qui est censée réaliser la propriété considérée. Afin d’établir un lien entre la (umap) et la structure du spectre E, nous faisons le détour par une étude générale de cette propriété dans le cadre des espaces de Banach séparables.

2.1.1 Amorce et queue d’un espace de Banach

Peter G. Casazza et Nigel J. Kalton ont découvert le critère suivant:

Proposition 2.1.1 Soit X un espace de Banach séparable. X a la (umap) si et seulement s’il existe une suite approximante {T_{k}} telle que

sup_{ϵ∈𝕊} ∥T_{k} + ϵ(Id − T_{k})∥_{ℒ(X)} ⟶_{k→∞}1.

Ceci exprime que la constante d’inconditionnalité entre l’amorce T_{k}X et la queue (Id − T_{k})X de l’espace X s’améliore asymptotiquement jusqu’à l’optimum pour k → ∞. La (umap) s’exprime de manière plus élémentaire encore si l’on choisit d’autres notions adaptées d’amorce et de queue. Nous proposons en particulier la définition suivante.

Définition 2.1.2 Soit τ une topologie d’espace vectoriel topologique sur X. X a la propriété (u(τ)) de τ-inconditionnalité si pour chaque x ∈ X et toute suite bornée {y_{j}} τ-nulle l’oscillation

osc_{ϵ∈𝕊} ∥x + ϵy_{j}∥_{X} = sup_{δ,ϵ∈𝕊}(∥x + ϵy_{j}∥ − ∥x + δy_{j}∥)

forme elle-même une suite nulle.

Nous avons alors le théorème suivant.

Théorème 2.1.3 Soit X un espace de Banach séparable de cotype fini avec la propriété (u(τ)). Si X admet une suite approximante {T_{k}} inconditionnelle et commutative telle que T_{k}x →^{τ} x uniformément sur la boule unité B_{X}, alors des combinaisons convexes successives {U_{j}} de {T_{k}} réalisent la (umap).

Esquisse de preuve. On construit ces combinaisons convexes successives par le biais de décompositions skipped blocking. En effet, la propriété (u(τ)) a l’effet suivant sur {T_{k}}. Pour chaque ε > 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 manière (1 + ε)-inconditionnelle. Soit n ≥ 1. Pour chaque j, 1 ≤ j ≤ n, la suite de blocs obtenue en sautant S_{kn+j} − S_{kn+j−1} pour k ≥ 0 est (1 + ε)-inconditionnelle. Il s’agit alors d’estimer la moyenne sur j de ces suites de blocs. On obtient une suite approximante et l’hypothèse de cotype fini permet de contrôler l’apport des blocs sautés. Alors X a la (umap) parce que n et ε sont arbitraires.

2.1.2 Amorce et queue en termes de spectre de Fourier

Lorsqu’on considère l’espace invariant par translation X_{E}, une amorce et une queue naturelle sont les espaces X_{F} et X_{E\G} pour F et G des sous-ensembles finis de E. Nous avons concrètement le lemme suivant.

Lemme 2.1.4 X_{E} a (u(τ_{f})), où τ_{f} est la topologie

f_{n} →^{τ_{f}} 0 ⟺ ∀k f̂_{n}(k) → 0

de convergence simple des coefficients de Fourier, si et seulement si E est blocinconditionnel dans X au sens suivant: quels que soient ε > 0 et F ⊆ E fini, il existe G ⊆ E fini tel que pour f ∈ B_{X_{F}} et g ∈ B_{X_{E\G}}

osc_{ϵ∈𝕊} ∥f + ϵg∥_{X} = sup_{δ,ϵ∈𝕊}(∥f + ϵg∥ − ∥f + δg∥) ≤ ε.

Le théorème 2.1.3 s’énonce donc ainsi dans ce contexte particulier.

Théorème 2.1.5 Soit E ⊆ ℤ et X un espace de Banach homogène 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 propriété d’approximation inconditionnelle et un cotype fini, alors X_{E} a la (umap). En particulier, on a

(i) Soit 1 < p < ∞. L_{E}^{p}(𝕋) a la (umap) si et seulement si E est bloc-inconditionnel dans L^{p}(𝕋). (ii) L_{E}^{1}(𝕋) a la (umap) si et seulement si L_{E}^{1}(𝕋) a la propriété d’approximation inconditionnelle et E est bloc-inconditionnel dans L^{1}(𝕋). (iii) Si E est bloc-inconditionnel dans 𝒞(𝕋) et E est un ensemble de Sidon, alors 𝒞_{E}(𝕋) a la (umap).

Donnons une application de ce théorème.

Proposition 2.1.6 Soit E = {n_{k}} ⊆ ℤ. Si n_{k+1}/n_{k} est un entier impair pour tout k, alors 𝒞_{E}(𝕋) a la (umap) réelle.

Preuve. Comme E est nécessairement un ensemble de Sidon, il suffit de vérifier que E est bloc-inconditionnel. Soient ε > 0 et F ⊆ E∩[−n, n]. Soit l tel que |n_{l}| ≥ πn/ε et G = {n_{1}, …, n_{l−1}}. Soit f ∈ B_{𝒞_{F}} et g ∈ B_{𝒞_{E\G}} . Alors g(t + π/n_{l}) = −g(t) par hypothèse et

|f(t + π/n_{l}) − f(t)| ≤ π/|n_{l}| · ∥f′∥_{∞} ≤ πn/|n_{l}| ≤ ε

par l’inégalité de Bernstein. Alors, pour un certain u ∈ 𝕋

∥f − g∥_{∞} = |f(u) + g(u + π/n_{l})|
≤ |f(u + π/n_{l}) + g(u + π/n_{l})| + ε
≤ ∥f + g∥_{∞} + ε.

Donc E est bloc-inconditionnel au sens réel. En particulier, soit la suite géométrique G = {3^{k}}. Alors 𝒞_{G}(𝕋) et 𝒞_{G∪−G}(𝕋) ont la (umap) réelle. Question 2.1.7 Qu’en est-il de la (umap) complexe et qu’en est-il de la suite géométrique G = {2^{k}} ?

2.2 Norme de multiplicateurs et conditions combinatoires

Nous proposons ici une méthode uniforme pour répondre aux questions 1.1.4, 1.1.5, 1.1.6, 1.1.9 et 1.1.10. En effet, les questions 1.1.4, 1.1.5 et 1.1.6 reviennent à évaluer l’oscillation de la norme

Θ(ϵ, a) = ∥ϵ_{0}a_{0} e_{r_{0}} + … + ϵ_{m}a_{m} e_{r_{m}} ∥_{X}.

La question 1.1.9 revient à évaluer l’oscillation de la norme

Ψ(ϵ, a) = Θ((\overbrace{1, …, 1}^{j}, \overbrace{ϵ, …, ϵ}^{m−j}), a)
= ∥a_{0} e_{r_{0}} + … + a_{j} e_{r_{j}} +ϵa_{j+1} e_{r_{j+1}} + … + ϵa_{m} e_{r_{m}} ∥_{X}

Par le théorème 2.1.5, la question 1.1.10 revient à étudier cette même expression dans le cas particulier où on fait un saut de grandeur arbitraire entre r_{j} et r_{j+1}. Dans le cas des espaces X = L^{p}(𝕋), p entier pair, ces normes sont des polynômes en ϵ, ϵ^{−1}, a et ā. Dans le cas des espaces X = L^{p}(𝕋), p non entier pair, elles s’expriment comme des séries. Il n’y a pas moyen d’exprimer ces normes comme fonction 𝒞^{∞} pour X = 𝒞(𝕋). Soit X = L^{p}(𝕋). Développons Θ(ϵ, a). Posons q_{i} = r_{i} − r_{0}. On peut supposer ϵ_{0} = 1 et a_{0} = 1. Nous utilisons la notation suivante:

\binom{x}{α} = \frac{x(x − 1) ⋯ (x − n + 1)}{α_{1}!α_{2}! …} pour α ∈ ℕ^{m} tel que ∑ α_{i} = n

Alors, si |a_{1}|, …, |a_{m}| < 1/m lorsque p n’est pas un entier pair et sans restriction sinon,

Θ(ϵ, a) = ∫ |∑_{n≥0} \binom{p/2}{n} (∑_{i=1}^{m} ϵ_{i}a_{i} e_{q_{i}})^{n}|^{2}
= ∫ |∑_{n≥0} \binom{p/2}{n} ∑_{{α:α_{1},...,α_{m}≥0 \atop α_{1}+...+α_{m}=n}} ({n \atop α})ϵ^{α}a^{α} e_{Σα_{i}q_{i}} |^{2}

= ∫ | ∑_{α∈ℕ^{m}} \binom{p/2}{α} ϵ^{α}a^{α} e_{Σα_{i}q_{i}} |^{2}

= ∑_{R∈ℛ} |∑_{α∈R} \binom{p/2}{α} ϵ^{α}a^{α}|^{2}
= ∑_{α∈ℕ^{m}} \binom{p/2}{α}^{2}|a|^{2α} + ∑_{{α≠β∈ℕ^{m} \atop α∼β}} \binom{p/2}{α} \binom{p/2}{β} ϵ^{α−β}a^{α}ā^{β}

où ℛ est la partition de ℕ^{m} induite par la relation d’équivalence

α ∼ β ⇔ ∑ α_{i}q_{i} = ∑ β_{i}q_{i}.

Nous pouvons répondre immédiatement aux questions 1.1.4 et 1.1.5 pour X = L^{p}(𝕋).

2.2.1 Question 1.1.4: suites basiques 1-inconditionnelles complexes

Soient r_{0}, … r_{m} sont choisis dans E, alors (4) doit être constante pour a ∈ {|z| < 1/m}^{m} et ϵ ∈ 𝕋^{m}. Cela veut dire que pour tous α ≠ β ∈ ℕ^{m},

∑ α_{i}q_{i} ≠ ∑ β_{i}q_{i} ou \binom{p/2}{α} \binom{p/2}{β} = 0.

Si p n’est pas un entier pair, alors \binom{p/2}{α} \binom{p/2}{β} ≠ 0 pour tous α, β ∈ ℕ^{m} et on a les relations arithmétiques suivantes sur q_{1}, q_{2}, 0:

\overbrace{q_{1} + … + q_{1}}^{|q_{2}|} = \overbrace{q_{2} + … + q_{2}}^{|q_{1}|} si q_{1}q_{2} > 0;
\overbrace{q_{1} + … + q_{1}}^{|q_{2}|} + \overbrace{q_{2} + … + q_{2}}^{|q_{1}|} = 0  sinon.

Il suffit donc de prendre α = (|q_{2}|, 0, …), β = (|q_{1}|, 0, …) et α = (|q_{2}|, |q_{1}|, 0, …), β = (0, …) respectivement pour conclure que {r_{0}, r_{1}, r_{2}} n’est pas une suite basique 1-inconditionnelle complexe dans L^{p}(𝕋) si p n’est pas un entier pair. Si p est un entier pair, \binom{p/2}{α} \binom{p/2}{β} = 0 si et seulement si

∑ α_{i} > p/2 ou ∑ β_{i} > p/2.

On obtient que E est une suite basique 1-inconditionnelle dans L^{p}(𝕋) si et seulement si E est “p-indépendant”, c’est-à-dire que ∑ α_{i}(r_{i} − r_{0}) ≠ ∑ β_{i}(r_{i} − r_{0}) pour tous r_{0}, …, r_{m} ∈ E et α ≠ β ∈ ℕ^{m} tels que ∑ α_{i}, ∑ β_{i} ≤ p/2. Cette condition est équivalente à: tout entier n ∈ ℤ s’écrit de manière au plus unique comme somme de p/2 éléments de E.

2.2.2 Question 1.1.5: suites basiques 1-inconditionnelles réelles

Les suites basiques 1-inconditionnelles réelles et complexes coïncident et la réponse à la question 1.1.5 est identique à la réponse à la question 1.1.4. En effet, dès qu’une relation arithmétique ∑(α_{i} − β_{i})q_{i} pèse sur E, on peut supposer que α_{i} − β_{i} est impair pour au moins un i en simplifiant la relation par le plus grand diviseur commun des α_{i} − β_{i}. Mais alors (4) n’est pas une fonction constante pour ϵ_{i} réel. Cette propriété est propre au tore 𝕋. En effet, par exemple la suite des fonctions de Rademacher est 1-inconditionnelle réelle dans 𝒞(𝔻^{∞}), alors que sa constante d’inconditionnalité complexe est π/2.

2.2.3 Question 1.1.6: suites basiques inconditionnelles métriques

On peut même tirer des conséquences utiles du calcul de (4) dans le cas presqu’isométrique. Il faut pour cela prendre la précaution suivante qui permet un passage à la limite. Soit 0 < ϱ < 1/m. Alors

{Θ: 𝕊^{m} × {|z| ≤ ϱ}^{m} → ℝ^{+} : q_{1}, …, q_{m} ∈ ℤ^{m}}

est un sous-ensemble relativement compact de 𝒞^{∞}(𝕊^{m} × {|z| ≤ ϱ}^{m}). Il en découle que si E est une suite basique inconditionnelle métrique, alors certains coefficients de (4) deviennent arbitrairement petits lorsque q_{1}, …, q_{m} sont choisis grands. Donnons deux conséquences de ce raisonnement.

Proposition 2.2.1 Soit E ⊆ ℤ.

(i) Soit p un entier pair. Si E est une suite basique inconditionnelle métrique réelle, alors E est en fait une suite basique 1-inconditionnelle complexe à un ensemble fini près. (ii) Si E est un ensemble de Sidon de constante asymptotiquement 1, alors

⟨ζ, E⟩ = sup_{G⊆E fini} inf{|ζ_{1}p_{1} + … + ζ_{m}p_{m}| : p_{1}, …, p_{m} ∈ E \ G distincts} > 0

pour tout m ≥ 1 et ζ ∈ ℤ^{∗}^{m}.

On peut exprimer cette dernière propriété en disant que la relation arithmétique ζ ne persiste pas sur E.

2.2.4 Question 1.1.10: propriété d’approximation inconditionnelle métrique

On peut appliquer la technique du paragraphe précédent en observant que si X_{E} a la (umap), alors

osc_{ϵ∈𝕊} Ψ(ϵ, a)⟶_{r_{j+1},...,r_{m}∈E→∞}0.

Définition 2.2.2 E a la propriété (𝒥_{n}) de bloc-indépendance si pour tout F ⊆ E fini il existe G ⊆ E fini tel que si un k ∈ ℤ admet deux représentations comme somme de n éléments de F ∪ (E \ G)

p_{1} + … + p_{n} = k = p′_{1} + … + p′_{n},

alors

{j : p_{j} ∈ F } et {j : p′_{j} ∈ F }

sont égaux (choix de signes complexe 𝕊 = 𝕋) ou de même parité (choix de signes réel 𝕊 = 𝔻).

Théorème 2.2.3 Soit E ⊆ ℤ.

(i) Si X = L^{p}(𝕋), p entier pair, alors L_{E}^{p}(𝕋) a la (umap) si et seulement si E satisfait (𝒥_{p/2}).

(ii) Si X = L^{p}(𝕋), p non entier pair, ou X = 𝒞(𝕋), alors X_{E} a la (umap) seulement si E satisfait

⟨ζ, E⟩ = sup_{G⊆E fini} inf{|ζ_{1}p_{1} + … + ζ_{m}p_{m}| : p_{1}, …, p_{m} ∈ E \ G distincts} > 0

pour tout m ≥ 1 et ζ ∈ ℤ^{∗}^{m} tel que ∑ ζ_{i} est non nul (cas complexe) ou impair (cas réel).

On obtient la hiérarchie suivante.

{𝒞_{E}(𝕋) a \atop (umap)} ⇒ L_{E}^{p}(𝕋) a (umap), p non entier pair ⇒ … ⇒ {L_{E}^{2n+2}(𝕋) \atop a (umap)} ⇒ {L_{E}^{2n}(𝕋) a \atop (umap)} ⇒ … ⇒ {L_{E}^{2}(𝕋) a \atop (umap).}

Nous pouvons répondre à la question 2.1.7. Soit G = {j^{k}} avec j ∈ ℤ \ {−1, 0, 1} et considérons ζ = (j, −1). Alors ⟨ζ, G⟩ = 0. Donc 𝒞_{G}(𝕋) n’a pas la (umap) complexe. 𝒞_{G}(𝕋) n’a pas la (umap) réelle si j est pair.

2.2.5 Deux exemples

À l’aide de nos conditions arithmétiques, nous sommes à même de prouver la proposition suivante.

Proposition 2.2.4 Soit σ > 1 et E la suite des parties entières de σ^{k}. Alors les assertions suivantes sont équivalentes.

(i) σ est un nombre transcendant. (ii) L_{E}^{p}(𝕋) a la (umap) complexe pour tout p entier pair. (iii) E est une suite basique inconditionnelle métrique dans chaque L^{p}(𝕋), p entier pair. (iv) Pour chaque m donné, la constante de Sidon des sous-ensembles à m éléments de queues de E est asymptotiquement 1.

Nous obtenons aussi la proposition suivante.

Proposition 2.2.5 Soit E la suite des bicarrés. L_{E}^{p}(𝕋) a la (umap) réelle seulement si p = 2 ou p = 4.

Preuve. E ne satisfait pas la propriété de bloc-indépendance (𝒥_{3}) réelle. En effet, Ramanujan a découvert l’égalité 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}.

2.3 Impact de la croissance du spectre

Nous démontrons de manière directe le résultat positif suivant.

Théorème 2.3.1 Soit E = {n_{k}} ⊆ ℤ tel que n_{k+1}/n_{k} → ∞. Alors la suite des projections associée à E réalise la (umap) complexe dans 𝒞_{E}(𝕋) et E est un ensemble de Sidon de constante asymptotiquement 1. Dans l’hypothèse où les rapports n_{k+1}/n_{k} sont tous entiers, la réciproque vaut.

Corollaire 2.3.2 Alors E est une suite basique inconditionnelle métrique dans tout espace de Banach homogène X de fonctions sur 𝕋. De plus, X_{E} a la (umap) complexe.

Esquisse de preuve. Nous prouvons concrètement que si n_{k+1}/n_{k} → ∞, alors quel que soit ε > 0 il existe l ≥ 1 tel que pour toute fonction f = ∑ a_{k} e_{n_{k}}

∥f∥_{∞} ≥ (1 − ε)(∥∑_{k≤l} a_{k} e_{n_{k}} ∥_{∞} + ∑_{k>l} |a_{k}|).  (4)

Cela revient à dire que la suite {π_{k}} de projections associée à la base E réalise la 1/(1 − ε)-(uap). Pour obtenir l’inégalité (4), on utilise une récurrence basée sur l’idée suivante. Soit u ∈ 𝕋 tel que ∥π_{k}f∥_{∞} = |π_{k}f(u)|. Il existe alors v ∈ 𝕋 tel que

|u − v| ≤ π/|n_{k+1}| et |π_{k}f(u) + a_{k+1} e_{n_{k+1}}(v)| = ∥π_{k}f∥_{∞} + |a_{k+1}|.

De plus, dans ce cas,

|π_{k}f(u) − π_{k}f(v)| ≤ |u − v| ∥π_{k}f′∥_{∞} ≤ π|n_{k}/n_{k+1}| ∥π_{k}f∥_{∞}.

En résumé, a_{k+1} e_{n_{k+1}} a le même argument que π_{k}f très près du maximum de |π_{k}f|, et π_{k}f varie peu. Mais alors

∥π_{k}f(t) + a_{k+1} e_{n_{k+1}} ∥_{∞} ≥ |π_{k}f(v) + a_{k+1} e_{n_{k+1}}(v)|
≥ ∥π_{k}f∥_{∞} + |a_{k+1}| − π|n_{k}/n_{k+1}|∥π_{k}f∥_{∞}
= (1 − π|n_{k}/n_{k+1}|)∥π_{k}f∥_{∞} + |a_{k+1}|.

On obtient (4) en réitérant cet argument. Notre technique donne d’ailleurs l’estimation suivante de la constante de Sidon des ensembles de Hadamard.

Corollaire 2.3.3 Soit E = \{n_{k}\} ⊆ ℤ et q > \sqrt{π^{2}/2 + 1}. Si |n_{k+1}| ≥ q|n_{k}|, alors la constante de Sidon de E est inférieure ou égale à 1 + π^{2}/(2q^{2} − 2 − π^{2}).

Nous prouvons que cette estimation est optimale au sens où l’ensemble E = {0, 1, q}, q ≥ 2, a pour constante d’inconditionnalité réelle dans 𝒞(𝕋)

(cos(π/(2q))^{−1} ≥ 1 + π^{2}/8 q^{−2}.

3 Suites basiques 1-inconditionnelles d’entrées de matrice

Dans ce chapitre, nous cherchons à répondre à la question 1.2.1. Nous fournissons une réponse complète dans le cas particulier des classes de Schatten S^{p} avec p entier pair. En effet, la technique présentée dans la section 2.2 peut être transférée du cadre des multiplicateurs de Fourier au cadre des multiplicateurs de Schur. Nous interprétons la condition combinatoire obtenue à l’aide d’objets combinatoires introduits ad hoc. Notre analyse aboutit au théorème suivant.

Théorème 3.1 Soit I ⊆ ℕ × ℕ.

(i) {e_{rc} : (r, c) ∈ I} est une suite basique 1-inconditionnelle réelle dans S^{p} exactement quand elle est 1-inconditionnelle complexe et même c.b. 1-inconditionnelle complexe. (ii) I satisfait ces trois propriétés exactement lorsque I est “matriciellement p/2indépendant”: deux points q, q′ ∈ I sont reliés par au plus un seul chemin sans retour sur le réseau ℕ × ℕ dont les au plus p/2 sommets sont dans I.

4 Constructions aléatoires à l’intérieur de suites lacunaires

Dans ce chapitre, nous fournissons une preuve nouvelle pour une construction aléatoire d’ensembles lacunaires par Yitzhak Katznelson qui appartient au folklore de l’analyse harmonique. Nous analysons et généralisons aussi la construction aléatoire d’ensembles équidistribués par Jean Bourgain. Cela nous permet d’établir le tableau 1.3.1 qui classe les propriétés de Rosenthal et Λ(p) pour tout p selon la croissance du spectre. Nous montrons alors que la démarche probabiliste suivie par Katznelson et Bourgain pour construire ces sousensembles de ℤ utilise seulement la croissance “arithmétique” et l’équidistribution de la suite des entiers ℤ. En fait, ces sous-ensembles peuvent être construits à l’intérieur de suites équidistribuées à croissance polynômiale. En particulier, le tableau 1.3.1 reste valable pour l’ensemble des sous-ensembles E d’une suite polynômiale, ainsi que de la suite des nombres premiers. Nous fournissons une réponse partielle à la question 1.3.3.

Théorème 4.1 Soit P une suite polynômiale ou la suite des nombres premiers. Alors il existe une sous-suite E de P qui est Λ(p) pour tout p alors qu’elle ne forme pas un ensemble de Rosenthal.

Le chapitre II correspond à l’article [73] publié dans Studia Mathematica sauf la section II.10.1, soumise au Bulletin of the London Mathematical Society. Le chapitre IV a été soumis aux Annales de l’Institut Fourier. Le chapitre III fait partie d’une recherche en cours.

Chapitre II

Metric unconditionality and Fourier analysis

1 Introduction

We study isometric and almost isometric counterparts to the following two properties of a separable Banach space Y:

(ubs) Y is the closed span of an unconditional basic sequence;

(uap) Y admits an unconditional finite dimensional expansion of the identity.

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 L^{p}(𝕋) (1 ≤ p < ∞) or 𝒞(𝕋). If {e^{int}}_{n∈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 Λ(q), q = p ∨ 2, and a Sidon set respectively. We raise the following question: what kind of lacunarity is needed to get the following stronger property:

(umbs) E is a metric unconditional basic sequence in X: for any ε > 0, one may lower its unconditionality constant to 1 + ε by removing a finite set from it.

In the case of 𝒞(𝕋), 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 (a.s. 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 ΔT_{k} = T_{k} − T_{k−1} the difference sequence of T_{k}. Following Rosenthal (see [27, §1]), we then say that Y has the unconditional approximation property ((uap) for short) if it admits an a.s. {T_{k}} such that for some C

∥∑_{k=1}^{n} ϵ_{k}ΔT_{k}∥_{ℒ(Y)} ≤ C  for all n and scalar ϵ_{k} with |ϵ_{k}| = 1.  (1)

By the uniform boundedness principle, (1) means exactly that ∑ ΔT_{k}y converges unconditionally for all y ∈ 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 [12, §3] ?

(umap) The span Y = X_{E} of E in X has the metric unconditional approximation property: for any ε > 0, one may lower the constant C in (1) to 1 + ε by choosing an adequate a.s. {T_{k}}.

Several kinds of metric, i. e. almost isometric properties have been investigated in the last decade (see [38]). There is a common feature to these notions since Kalton’s [47]: 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”.

Property (umap) has been studied by Li [58] for X = 𝒞(𝕋). 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 𝒞_{E}(𝕋) has (umap) for all (umbs) E in 𝒞(𝕋): for sets of the latter kind, the natural sequence of projections realizes (uap) in 𝒞_{E}(𝕋), but we do not know whether it achieves (umap)).

In fact, both problems lead to strong arithmetical conditions on E that are somewhat complementary to the property of quasi-independence (see [77, §3]). In order to obtain them, we apply Forelli’s [28, Prop. 2] and Plotkin’s [79, Th. 1.4] techniques in the study of isometric operators on L^{p}: see Theorem 2.4.2 and Lemma 7.1.4. 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 6.2.3. As in Forelli’s and Plotkin’s results, we obtain that the spaces X = L^{p}(𝕋) with p an even integer play a special rôle. For instance, they are the only spaces which admit 1-unconditional basic sequences E ⊆ ℤ with more than two elements: see Proposition 2.2.1.

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 ∈ ℤ, Rosenblatt’s [83] strong mixing coefficient

sup{|m[A ∩ B] − m[A]m[B]| : A ∈ σ(e^{ikt}) and B ∈ σ(e^{ilt})}

has its maximum value, 1/4. But lacunarity of E enhances their independence in several weaker senses (see [3]). Properties (umap) and (umbs) can be seen as an expression of almost independence of elements of E in the “additive sense”, i. e. when appearing in sums. We show their relationship to the notions of pseudoindependence (see [72, §4.2]) and almost i.i.d. sequences (see [2]).

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 L^{2n}(𝕋), 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 L^{p}(𝕋), p not an even integer, and 𝒞(𝕋), these arithmetical conditions are infinitely many and become much more coercive. In particular, if our properties are satisfied in 𝒞(𝕋), then they are satisfied in all spaces L^{p}(𝕋), 1 ≤ p < ∞.

We now turn to a detailed discussion of our results: in Section 2, we first characterize the sets E and values p such that E is a 1-unconditional basic sequence in L^{p}(𝕋) (Prop. 2.2.1). Then we show how to treat similarly the almost isometric case and obtain a range of arithmetical conditions (ℐ_{n}) on E (Th. 2.4.2). 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 L^{p}(𝕋) spaces with p not an even integer and 𝒞(𝕋), but single out L^{p}(𝕋) with p an even integer: this property does not “interpolate”. This is similar to the phenomena of equimeasurability (see [55, introduction]) and 𝒞^{∞}-smoothness of norms (see [14, Chapter V]). These facts may also be appreciated from the point of view of natural renormings of the Hilbert space L_{E}^{2}(𝕋).

In Section 3, 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. 3.3.1).

In Section 4, 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 and Rosenthal [10] gives a canonical way to construct an a.s. that realizes (umap) (Th. 4.3.1).

In Section 5, we introduce the p-additive approximation property ℓ_{p}-(ap) and its metric counterpart, ℓ_{p}-(map). It may be described as simply as (umap). Then we connect ℓ_{p}-(map) with the work of Godefroy, Kalton, Li and Werner [48, 32] on subspaces of L^{p} which are almost isometric to ℓ_{p}.

Section 6 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 6.2.3.

In Section 7, we proceed as in Section 2 to obtain a range of arithmetical conditions (𝒥_{n}) for (umap) and metric unconditional (fdd) (Th. 7.2.1 and Prop. 7.2.4). These conditions are similar to (ℐ_{n}), but are decidedly weaker: see Proposition 8.1.2(i). This time, real and complex unconditionality differ; again spaces L^{p}(𝕋) with even p are singled out.

In Section 8, we continue the arithmetical investigation begun in Section 3 with property (𝒥_{n}) and obtain many examples for the 1-unconditional and the metric unconditional approximation property.

However, the main result of Section 9, Theorem 9.3.1, 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 𝒞_{{3^{k}}}(𝕋) has real (umap) and that this is due to the oddness of 3 (Prop. 9.1.1). A sharp estimate of the Sidon constant of Hadamard sets is obtained as a byproduct (Cor. 10.2.1). We compute the Sidon constant of sets with three elements (Th. 10.1.5).

Section 11 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 [58]: for X = 𝒞(𝕋) and for X = L^{p}(𝕋), p ≠ 2, 4, the maximal density d^{∗} of E is zero if X_{E} has (umap) (Prop. 11.2). For X = L^{4}(𝕋), our technique falls short of the expected result: we just know that if L_{E∪{a}}^{4}(𝕋) has (umap) for every a ∈ ℤ, then d^{∗}(E) = 0.

Section 12 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 [2]. We note further that the arithmetical property (ℐ_{∞}) of Section 2 is equivalent to Murai’s [72, §4.2] property of pseudo-independence.

In Section 13, we collect our results on metric unconditional basic sequences of characters and (umap) in translation invariant spaces. We conclude with open questions.

Notation and definitions Sections 2, 6, 7 and 9 will take place in the following framework. (𝕋, dm) denotes the compact abelian group {z ∈ ℂ : |z| = 1} endowed with its Haar measure dm; m[A] is the measure of a subset A ⊆ 𝕋. Let 𝔻 = {−1, 1}. 𝕊 will denote either the complex (𝕊 = 𝕋) or real (𝕊 = 𝔻) choice of signs. For a real function f on 𝕊, the oscillation of f is

osc_{ϵ∈𝕊} f(ϵ) = sup_{ϵ∈𝕊} f(ϵ) − inf_{ϵ∈𝕊} f(ϵ).

We shall study homogeneous Banach spaces X of functions on 𝕋 [50, Chapter I.2], and especially the peculiar behaviour of the following ones: L^{p}(𝕋) (1 ≤ p < ∞), the space of p-integrable functions with the norm ∥f∥_{p} = (∫ |f|^{p}dm)^{1/p}, and 𝒞(𝕋), the space of continuous functions with the norm ∥f∥_{∞} = max{|f(t)| : t ∈ 𝕋}. ℳ(𝕋) is the dual of 𝒞(𝕋) realized as Radon measures on 𝕋. The dual group {e_{n}: z ↦ z^{n} : n ∈ ℤ} of 𝕋 is identified with ℤ. We write B for the cardinal of a set B. For a not necessarily increasing sequence E = {n_{k}}_{k≥1} ⊆ ℤ, let 𝒫_{E}(𝕋) 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 ∈ X_{E} and n ∉ E, f̂(n) = ∫ f(t) e_{−n}(t)dm(t) = 0. X_{E} is also the closure of 𝒫_{E}(𝕋) in homogeneous X [50, Th. 2.12]. Denote by π_{k} : X_{E} → X_{E} the orthogonal projection onto X_{{n_{1},...,n_{k}}}. It is given by

π_{k}(f) = f̂(n_{1}) e_{n_{1}} + … + f̂(n_{k}) e_{n_{k}} .

Then the π_{k} commute. They form an a.s. for X_{E} if and only if E is a basic sequence. For a finite or cofinite F ⊆ E, π_{F} is similarly the orthogonal projection of X_{E} onto X_{F}. Sections 4 and 5 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 ΔU_{k} = U_{k} − U_{k−1} (where U_{0} = 0). The functional notions of (ubs), (umbs) are defined in 2.1.1. The functional notions of a.s., (uap) and (umap) are defined in 4.1.1. Properties ℓ_{p}-(ap) and ℓ_{p}-(map) are defined in 5.1.1. The functional property (𝒰) of block unconditionality is defined in 6.2.1. The sets of arithmetical relations Z^{m} and Z_{n}^{m} are defined before 2.2.1. The arithmetical properties (ℐ_{n}) of almost independence and (𝒥_{n}) of block independence are defined in 2.4.1 and 7.1.2 respectively. The pairing ⟨ζ, E⟩ is defined before 3.1.1.

2 Metric unconditional basic sequences of characters (umbs)

2.1 Definitions. Isomorphic case

We start with the definition of metric unconditional basic sequences ((umbs) for short). 𝕊 = 𝕋 = {ϵ ∈ ℂ : |ϵ| = 1} (vs. 𝕊 = 𝔻 = {−1, 1}) is the complex (vs. real) choice of signs.

Definition 2.1.1 Let E ⊆ ℤ and X be a homogeneous Banach space on 𝕋.

(i) [49] E is an unconditional basic sequence (ubs) in X if there is a constant C such that

∥∑_{q∈G} ϵ_{q}a_{q} e_{q}∥_{X} ≤ C∥∑_{q∈G} a_{q} e_{q}∥_{X}  (2)

for all finite subsets G ⊆ E, coefficients a_{q} ∈ ℂ and signs ϵ_{q} ∈ 𝕋 (vs. ϵ_{q} ∈ 𝔻). The infimum of such C is the complex (vs. real) unconditionality constant of E in X. If C = 1 works, then E is a complex (vs. real) 1-(ubs) in X. (ii) E is a complex (vs. real) metric unconditional basic sequence (umbs) in X if for each ε > 0 there is a finite set F such that the complex (vs. real) unconditionality constant of E \ F is less than 1 + ε.

Note that ℤ itself is an (ubs) in L^{p}(𝕋) if and only if p = 2 by Khinchin’s inequality. The same holds in the framework of the Cantor group 𝔻^{∞} and its dual group of Walsh functions: their common feature with the e_{n} is that their modulus is everywhere equal to 1 (see [54]). The following facts are folklore.

Proposition 2.1.2 Let Y be a Banach space.

(i) If ∥∑ ϵ_{k}y_{k}∥_{Y} ≤ C∥∑ y_{k}∥_{Y} for all ϵ_{k} ∈ 𝕋 (vs. ϵ_{k} ∈ 𝔻), then this holds automatically for all complex (vs. real) ϵ_{k} with |ϵ_{k}| ≤ 1. (ii) Real and complex unconditionality are isomorphically π/2-equivalent.

Proof. (i) follows by convexity. (ii) Let us use the fact that the complex unconditionality constant of the Rademacher sequence is π/2 [89]:

sup_{δ_{k}∈𝕋}∥∑ δ_{k}y_{k}∥_{Y} = sup_{y^{∗}∈Y^{∗}} sup_{δ_{k}∈𝕋} sup_{ϵ_{k}=±1}|∑ δ_{k}⟨y^{∗}, y_{k}⟩ϵ_{k}|
≤ π/2 sup_{y^{∗}∈Y^{∗}} sup_{ϵ_{k}=±1}|∑⟨y^{∗}, y_{k}⟩ϵ_{k}| = π/2 sup_{ϵ_{k}=±1}∥∑ ϵ_{k}y_{k}∥_{Y} .

Taking the Rademacher sequence in 𝒞(𝔻^{∞}), we see that π/2 is optimal.

In fact, if (2) holds, then E is a basis of its span in X, which is X_{E} [50, Th. 2.12]. We have the following relationship between the unconditionality constants of E in 𝒞(𝕋) and in a homogeneous Banach space X on 𝕋.

Proposition 2.1.3 Let E ⊆ ℤ and X be a homogeneous Banach space on 𝕋.

(i) The complex (vs. real) unconditionality constant of E in X is at most the complex (vs. real) unconditionality constant of E in 𝒞(𝕋). (ii) If E is a (ubs) (vs. 1-(ubs), (umbs)) in 𝒞(𝕋), then E is a (ubs) (vs. 1-(ubs), (umbs)) in X.

This follows from the well-known (see e.g. [39])

Lemma 2.1.4 Let E ⊆ ℤ and X be a homogeneous Banach space on 𝕋. Let T be a multiplier on 𝒞_{E}(𝕋). Then T is also a multiplier on X_{E} and

∥T ∥_{ℒ(X_{E})} ≤ ∥T ∥_{ℒ(𝒞_{E})} .

Proof. The linear functional f ↦ T f(0) on 𝒞_{E}(𝕋) extends to a measure μ ∈ ℳ(𝕋) such that ∥μ∥_{ℳ} = ∥T ∥_{ℒ(𝒞_{E})} . Let μ̌(t) = μ(−t). Then T f = μ̌ ∗ f for f ∈ 𝒫_{E}(𝕋) and

∥T ∥_{ℒ(X_{E})} ≤ ∥μ̌∥_{ℳ} = ∥T ∥_{ℒ(𝒞_{E})} .

Question 2.1.5 There is no interpolation theorem for such relative multipliers. The forthcoming Theorem 2.4.2 shows that there can be no metric interpolation. Is it possible that one cannot interpolate multipliers at all between L_{E}^{p}(𝕋) and L_{E}^{q}(𝕋) ?

Note that conversely, [29] furnishes the example of an E ⊆ ℤ such that the π_{k} are uniformly bounded on L_{E}^{1}(𝕋) but not on 𝒞_{E}(𝕋). It is known that E is an (ubs) in 𝒞(𝕋) (vs. in L^{p}(𝕋)) if and only if it is a Sidon (vs. Λ(2 ∨ p)) set. To see this, let us recall the relevant definitions.

Definition 2.1.6 Let E ⊆ ℤ.

(i) [44] E is a Sidon set if there is a constant C such that

∑_{q∈G} |a_{q}| ≤ C∥∑_{q∈G} a_{q} e_{q}∥_{∞} for all finite G ⊆ E and a_{q} ∈ ℂ.

The infimum of such C is E’s Sidon constant. (ii) [86, Def. 1.5] Let p > 1. E is a Λ(p) set if there is a constant C such that ∥f∥_{p} ≤ C∥f∥_{1} for f ∈ 𝒫_{E}(𝕋).

In fact, the Sidon constant of E is the complex unconditionality constant of E in 𝒞(𝕋). Thus E is a complex (umbs) in 𝒞(𝕋) 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 Λ(2 ∨ p) set if and only if L_{E}^{p}(𝕋) = L_{E}^{2}(𝕋). Therefore Λ(2 ∨ p) sets are (ubs) in L^{p}(𝕋). Conversely, if E is an (ubs) in L^{p}(𝕋), then by Khinchin’s inequality

∥∑_{q∈G} a_{q} e_{q}∥_{p}^{p} ≈ average∥∑_{q∈G} ±a_{q} e_{q}∥_{p}^{p} ≈ (∑_{q∈G} |a_{q}|^{2})^{p/2} = ∥∑_{q∈G} a_{q} e_{q}∥_{2}^{p}

for all finite G ⊆ E (see [86, proof of Th. 3.1]). This shows also that the Λ(p) set constant and the unconditionality constant in L^{p}(𝕋) are connected via the constants in Khinchin’s inequality; whereas Sidon sets have their unconditionality constant in L^{p}(𝕋) uniformly bounded, the Λ(p) set constant of infinite sets grows at least like \sqrt{p} [86, Th. 3.4].

2.2 Isometric case: 1-unconditional basic sequences of characters

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: let A_{n} = {α = {α_{p}}_{p≥1} : α_{p} ∈ ℕ & α_{1} + α_{2} + … = n}. If α ∈ A_{n}, all but a finite number of the α_{p} vanish and the multinomial number

({n \atop α}) = \frac{n!}{α_{1}!α_{2}! …}

is well defined. Let A_{n}^{m} = {α ∈ A_{n} : α_{p} = 0 for p > m}. Note that A_{n}^{m} is finite. We call E n-independent 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

∑ α_{i}p_{i} = ∑ β_{i}p_{i} ⇒ α = β for α, β ∈ A_{n}^{m} and distinct p_{1}, …, p_{m} ∈ E.

This notion is studied in [17] where it is called birelation. In Rudin’s [86, §1.6(b)] notation, the number r_{n}(E; k) of representations of k ∈ 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

Z^{m} = {ζ ∈ ℤ^{∗}^{m} : ζ_{1} + … + ζ_{m} = 0} & Z_{n}^{m} = {ζ ∈ Z^{m} : |ζ_{1}| + … + |ζ_{m}| ≤ 2n}.

Note that Z_{n}^{m} is finite, and void if m > 2n. Then E is n-independent if and only if

∑ ζ_{i}p_{i} ≠ 0 for all ζ ∈ Z_{n}^{m} and distinct p_{1}, …, p_{m} ∈ E.

We shall prefer to treat arithmetical relations in terms of Z_{n}^{m} rather than A_{n}^{m}.

Proposition 2.2.1 Let E ⊆ ℤ.

(i) E is a complex 1-(ubs) in L^{p}(𝕋), p not an even integer, or in 𝒞(𝕋), if and only if E has at most two elements. (ii) If p is an even integer, then E is a complex 1-(ubs) in L^{p}(𝕋) 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 L^{p}(𝕋) or the complex unconditionality constant of E in L^{p}(𝕋) is at least C_{p}.

Proof. (i) By Proposition 2.1.3(ii), if E is not a complex 1-(ubs) in some L^{p}(𝕋), then neither in 𝒞(𝕋). Let p be not an even integer. We may suppose 0 ∈ E; let {0, k, l} ⊆ E. If we had ∥1+μa e_{k} +νb e_{l} ∥_{p} = ∥1+a e_{k} +b e_{l} ∥_{p} for all μ, ν ∈ 𝕋, then

∫ |1 + a e_{k} +b e_{l} |^{p}dm = ∫ |1 + μa e_{k} +νb e_{l} |^{p}dm(μ)dm(ν)dm
= ∫ |1 + μa + νb|^{p}dm(μ)dm(ν).

Denoting by θ_{i}: (ϵ_{1}, ϵ_{2}) ↦ ϵ_{i} the projections of 𝕋^{2} onto 𝕋, this would mean that ∥1 + a e_{k} +b e_{l} ∥_{p} = ∥1 + aθ_{1} + bθ_{2}∥_{L^{p}(𝕋^{2})} for all a, b ∈ ℂ. By [87, Th. I], (e_{k}, e_{l}) and (θ_{1}, θ_{2}) would have the same distribution. This is false, since θ_{1} and θ_{2} are independent random variables while e_{k} and e_{l} are not. (ii) Let q_{1}, …, q_{m} ∈ E be distinct and ϵ_{1}, …, ϵ_{m} ∈ 𝕋. By the multinomial formula for the power p/2 and Bessel–Parseval’s formula, we get

∥∑_{i=1}^{m} ϵ_{i}a_{i} e_{q_{i}} ∥_{p}^{p} = ∫ | ∑_{α∈A_{p/2}^{m}} \binom{p/2}{α} ∏_{i=1}^{m}(ϵ_{i}a_{i})^{α_{i}} e_{Σα_{i}q_{i}} |^{2}dm

= ∑_{A∈ℛ_{q}} |∑_{α∈A} \binom{p/2}{α} ∏_{i=1}^{m}(ϵ_{i}a_{i})^{α_{i}} |^{2}
= ∑_{α∈A_{p/2}^{m}} \binom{p/2}{α}^{2} ∏_{i=1}^{m} |a_{i}|^{2α_{i}} + ∑_{{α≠β∈A_{p/2}^{m} \atop α∼β}} \binom{p/2}{α} \binom{p/2}{β} ∏_{i=1}^{m} ϵ_{i}^{α_{i}−β_{i}}a_{i}^{α_{i}} \overline{a_{i}}^{β_{i}},

where ℛ_{q} is the partition of A_{p/2}^{m} induced by the equivalence relation α ∼ β ⇔ ∑ α_{i}q_{i} = ∑ β_{i}q_{i}. If E is p/2-independent, the second sum in (3) is void and E is a 1-(ubs).

Furthermore, suppose E is not p/2-independent and let q_{1}, …, q_{m} ∈ E be a minimal set of distinct elements of E such that there are α, β ∈ A_{p/2}^{m} with α ∼ β. Then m ≤ p. Take a_{i} = 1 in the former computation: then the clearly nonzero oscillation of (3) for ϵ_{1}, …, ϵ_{m} ∈ 𝕋 does only depend on ℛ_{q} and thus is finitely valued. This yields C_{p}.

Example 2.2.2 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} or 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 ϱ > 0. Then

∫ | e_{q_{1}} +ϱ e_{q_{2}} +ϵϱ e_{q_{3}} |^{4}dm = 1 + 6ϱ^{4} + 4ϱ^{2}(2 + ℜϵ).

Taking ϵ = −1 and ϵ = 1, ϱ = 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} ≠ 2q_{3} and q_{2} + q_{3} ≠ 2q_{4}: otherwise the first case occurs. Then

∫ | e_{q_{1}} + e_{q_{2}} + e_{q_{3}} +ϵ e_{q_{4}} |^{4}dm = 28 + 8ℜϵ.

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} ≈ 1.16 is the optimal choice for the constant in Proposition 2.2.1(ii).

Remark 2.2.3 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. 2.2.1 (i) for 𝒞(𝕋).

Remark 2.2.4 In fact the conclusion in (ii) holds also if we assume that E is just a real 1-(ubs). If we have some arithmetical relation α ∼ β, we may assume that α_{i} −β_{i} is odd for one i at least. Indeed, we may simplify all α_{i} −β_{i} by their greatest common divisor and this yields another arithmetical relation ∑(α′_{i} −β′_{i})q_{i} = 0. But then the oscillation of (3) is again clearly nonzero for ϵ_{1}, …, ϵ_{m} ∈ 𝔻.

Remark 2.2.5 We shall see in Remark 2.3.3 that (i) also holds in the real setting. This is a property of 𝕋 and fails for the Cantor group 𝔻^{∞}: the Rademacher sequence forms a real 1-(ubs) in 𝒞(𝔻^{∞}) but is clearly not complex 1-unconditional in any space L^{p}(𝔻^{∞}), p ≠ 2: see Section 12 and [89].

Question 2.2.6 There are nevertheless subspaces of L^{p}(𝕋), p not an even integer, and 𝒞(𝕋) with 1-unconditional bases, like sequences of functions with disjoint support. What about spaces L_{E}^{p}(𝕋) and 𝒞_{E}(𝕋), in particular when E is finite ? Are there 1-unconditional bases that do not consist of characters ?

Remark 2.2.7 For each even integer p ≥ 4, there are p/2-independent sets that are not Λ(p + ε) for any ε > 0: such maximal Λ(p) sets are constructed in [86].

2.3 Almost isometric case. A computation

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 2.2.1(i), we used the fact that the e_{n}, seen as random variables, are dependent: the 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. The contrast with the other L^{p} norms becomes clear when we try to compute explicitly an expression of type ∥∑ ϵ_{q}a_{q} e_{q}∥_{p} for any p ∈ [1, ∞[. This sort of seemingly brutal computation has been applied successfully in [28, Prop. 2] and [79, Th. 1.4] to study isometric operators on L^{p}, p not an even integer. We now undertake this tedious computation as preparatory work for Theorem 2.4.2, Lemma 7.1.4 and Proposition 7.2.4. Let us fix some more notation: for x ∈ ℝ and α ∈ A_{n}, put

({x \atop α}) = ({x \atop n})({n \atop α}).

This generalized multinomial number is nonzero if and only if x ≥ n or x ∉ ℕ.

Computational lemma 2.3.1 Let 𝕊 = 𝕋 or 𝕊 = 𝔻 in the complex and real case respectively. Let 1 ≤ p < ∞ and m ≥ 1. Put

φ_{q}(ϵ, z, t) = |1 + ∑_{i=1}^{m} ϵ_{i}z_{i} e_{q_{i}}(t)|^{p} , Φ_{q}(ϵ, z) = ∫ φ_{q}(ϵ, z, t) dm(t)

for q = (q_{1}, …, q_{m}) ∈ ℤ^{m}, ϵ = (ϵ_{1}, …, ϵ_{m}) ∈ 𝕊^{m} and z = (z_{1}, …, z_{m}) ∈ D^{m}, where D is the disc {|w| ≤ ϱ} ⊆ ℂ for some 0 < ϱ < 1/m. Define the equivalence relation α ∼ β ⇔ ∑ α_{i}q_{i} = ∑ β_{i}q_{i}. Then

Φ_{q}(ϵ, z) = ∑_{α∈ℕ^{m}} \binom{p/2}{α}^{2} ∏ |z_{i}|^{2α_{i}} + ∑_{{α≠β∈ℕ^{m} \atop α∼β}} \binom{p/2}{α} \binom{p/2}{β} ∏ z_{i}^{α_{i}} \overline{z_{i}}^{β_{i}}ϵ_{i}^{α_{i}−β_{i}}.

Furthermore, {Φ_{q} : q ∈ ℤ^{m}} is a relatively compact subset of 𝒞^{∞}(𝕊^{m} × D^{m}).

Proof. The function Φ_{q} is infinitely differentiable on the compact set 𝕊^{m} × D^{m}. Furthermore the family {Φ_{q} : q_{1}, …, q_{m} ∈ ℤ} is bounded in 𝒞^{∞}(𝕊^{m} × D^{m}) and henceforth relatively compact by Montel’s theorem. Let us compute φ_{q}. By the expansion of the function (1 + w)^{p/2}, analytic on the unit disc, and the multinomial formula, we have

φ_{q}(ϵ, z) = |∑_{a≥0} \binom{p/2}{a} (∑_{i=1}^{m} ϵ_{i}z_{i} e_{q_{i}})^{a}|^{2}
= |∑_{a≥0} \binom{p/2}{a} ∑_{α∈A_{a}^{m}} ({a \atop α}) ∏(ϵ_{i}z_{i})^{α_{i}} e_{Σα_{i}q_{i}} |^{2}
= | ∑_{α∈ℕ^{m}} \binom{p/2}{α} ∏(ϵ_{i}z_{i})^{α_{i}} e_{Σα_{i}q_{i}} |^{2}.

Let ℛ_{q} be the partition of ℕ^{m} induced by ∼. Then, by Bessel–Parseval’s formula

Φ_{q}(ϵ, z) = ∑_{A∈ℛ_{q}} |∑_{α∈A} \binom{p/2}{α} ∏(ϵ_{i}z_{i})^{α_{i}} |^{2}

and this gives (3) by expanding the modulus.

Remark 2.3.2 If m ≥ 2, this expansion has a finite number of terms if and only if p is an even integer: then and only then \binom{p/2}{α} = 0 for ∑ α_{i} > p/2, whereas ℛ_{q} contains clearly some class with two elements and thus an infinity thereof. For example, we have the following arithmetical relation on q_{1}, q_{2} or q_{1}, q_{2}, 0 respectively:

\overbrace{q_{1} + … + q_{1}}^{|q_{2}|} = \overbrace{q_{2} + … + q_{2}}^{|q_{1}|} if sgn q_{1} = sgn q_{2};

\overbrace{q_{1} + … + q_{1}}^{|q_{2}|} + \overbrace{q_{2} + … + q_{2}}^{|q_{1}|} = 0 if not.

Remark 2.3.3 This shows that Proposition 2.2.1(i) holds also in the real setting: we may suppose that 0 ∈ E; take m = 2 and choose q_{1}, q_{2} ∈ E. One of the two relations in Remark 2.3.2 yields an arithmetical relation on E with at least one odd coefficient, as done in Remark 2.2.4. But then (3) contains terms nonconstant in ϵ_{1} ∈ 𝔻 or in ϵ_{2} ∈ 𝔻 and thus E cannot be a real 1-unconditional basic sequence in L^{p}(𝕋).

We return to our computation.

Computational lemma 2.3.4 Let r = (r_{0}, …, r_{m}) ∈ E^{m+1} and put q_{i} = r_{i} − r_{0} (1 ≤ i ≤ m). Define

Θ_{r}(ϵ, z) = ∫ |e_{r_{0}} + ∑_{i=1}^{m} ϵ_{i}z_{i} e_{r_{i}} |^{p} = Φ_{q}(ϵ, z)  (3)

Let ζ_{0}, …, ζ_{m} ∈ ℤ^{∗} and

(γ_{i}, δ_{i}) = (−ζ_{i} ∨ 0, ζ_{i} ∨ 0)  (1 ≤ i ≤ m).  (4)

If the arithmetical relation

ζ_{0}r_{0} + … + ζ_{m}r_{m} = 0  while  ζ_{0} + … + ζ_{m} = 0  (5)

holds, then the coefficient of ∏ z_{i}^{γ_{i}} \overline{z_{i}}^{δ_{i}}ϵ_{i}^{γ_{i}−δ_{i}} in (3) is \binom{p/2}{γ} \binom{p/2}{δ} and thus independent of r. If ∑ |ζ_{i}| ≤ p or p is not an even integer, this coefficient is nonzero.

Proof. We have δ_{i} −γ_{i} = ζ_{i}, ∑ γ_{i} −∑ δ_{i} = ζ_{0} and ∑ γ_{i} +∑ δ_{i} = |ζ_{1}|+…+|ζ_{m}|, so that ∑ γ_{i} ∨ ∑ δ_{i} = \frac{1}{2} ∑ |ζ_{i}|. Moreover ∑(δ_{i} − γ_{i})q_{i} = ∑ ζ_{i}r_{i} = 0, so that γ ∼ δ.

2.4 Almost independent sets of integers. Main theorem

The Computational lemmas suggest the following definition.

Definition 2.4.1 Let E ⊆ ℤ.

(i) E enjoys the property (ℐ_{n}) of almost n-independence provided there is a finite subset G ⊆ E such that E \ G is n-independent, i. e. ζ_{1}r_{1} + … + ζ_{m}r_{m} ≠ 0 for all ζ ∈ Z_{n}^{m} and r_{1}, …, r_{m} ∈ E \ G. (ii) E enjoys exactly (ℐ_{n}) if furthermore it fails (ℐ_{n+1}). (iii) E enjoys (ℐ_{∞}) if it enjoys (ℐ_{n}) for all n, i. e. for any ζ ∈ Z^{m} there is a finite set G such that ζ_{1}r_{1} + … + ζ_{m}r_{m} ≠ 0 for r_{1}, …, r_{m} ∈ E \ G.

Note that property (ℐ_{1}) is void and that (ℐ_{n+1}) ⇒ (ℐ_{n}). This property is also stable under unions with a finite set. The preceding computations yield

Theorem 2.4.2 Let E = {n_{k}} ⊆ ℤ and 1 ≤ p < ∞.

(i) Suppose p is an even integer. Then E is a real, and at the same times complex, (umbs) in L^{p}(𝕋) if and only if E enjoys (ℐ_{p/2}). If (ℐ_{p/2}) holds, there is in fact a finite G ⊆ E such that E \ G is a 1-(ubs) in L^{p}(𝕋). (ii) If p is not an even integer and E is a real or complex (umbs) in L^{p}(𝕋), then E enjoys (ℐ_{∞}).

Proof. Sufficiency in (i) follows directly from Proposition 2.2.1: if E \ G is p/2independent, then E \ G is a real and complex 1-(ubs). Let us prove the necessity of the arithmetical property. We keep the notation of Computational lemmas 2.3.1 and 2.3.4. Assume E fails (ℐ_{n}) and let ζ_{0}, …, ζ_{m} ∈ ℤ^{∗} with ∑ ζ_{i} = 0 and ∑ |ζ_{i}| ≤ 2n such that for each l ≥ 1 there are distinct r_{0}^{l}, …, r_{m}^{l} ∈ E \ {n_{1}, …, n_{l}} with ζ_{0}r_{0}^{l} + … + ζ_{m}r_{m}^{l} = 0. One may furthermore assume that at least one of the ζ_{i} is not even. Assume E is a (umbs) in L^{p}(𝕋). Then the oscillation of Θ_{r} in (3) satisfies

osc_{ϵ∈𝕊^{m}} Θ_{r^{l}}(ϵ, z) ⟶_{l→∞} 0  (6)

for each z ∈ D^{m}. We may assume that the sequence of functions Θ_{r^{l}} converges in 𝒞^{∞}(𝕊^{m} × D^{m}) to a function Θ. Then by (6), Θ(ϵ, z) is constant in ϵ for each z ∈ D^{m}: in particular, its coefficient of ∏ z_{i}^{γ_{i}} \overline{z_{i}}^{δ_{i}}ϵ_{i}^{γ_{i}−δ_{i}} is zero. (Note that at least one of the γ_{i} − δ_{i} is not even). This is impossible by Computational lemma 2.3.4 if p is either not an even integer or if p ≥ 2n.

Corollary 2.4.3 Let E ⊆ ℤ. If E is a (umbs) in 𝒞(𝕋), that is E’s Sidon constant is asymptotically 1, then E enjoys (ℐ_{∞}). The converse does not hold.

Proof. Necessity follows from Theorem 2.4.2 and Proposition 2.1.3(ii). There is a counterexample to the converse in [86, Th. 4.11]: Rudin constructs a set E that enjoys (ℐ_{∞}) while E is not even a Sidon set.

For p an even integer, Sections 3 and 11 will provide various examples of (umbs) in L^{p}(𝕋). Proposition 9.2.1 gives a general growth condition on E under which it is an (umbs). As we do not know any partial converse to Theorem 2.4.2(ii) and Corollary 2.4.3, the sole known examples of (umbs) in L^{p}(𝕋), p not an even integer, and 𝒞(𝕋) are those given by Theorem 9.3.1. This theorem will therefore provide us with Sidon sets of constant asymptotically 1. Note, however, that Li [58, Th. 4] already constructed implicitly such a Sidon set by using Kronecker’s theorem.

3 Examples of metric unconditional basic sequences

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.

3.1 General considerations

The quantity

⟨ζ, E⟩ = sup_{G⊆E finite} inf{|ζ_{1}p_{1} + … + ζ_{m}p_{m}| : p_{1}, …, p_{m} ∈ E \ G distinct}
= lim_{l→∞} inf{|ζ_{1}p_{1} + … + ζ_{m}p_{m}| : p_{1}, …, p_{m} ∈ {n_{l}, n_{l+1}, …} distinct},

where {n_{k}} = E, plays a key rôle. We have

Proposition 3.1.1 Let E = {n_{k}} ⊆ ℤ.

(i) E enjoys (ℐ_{n}) if and only if ⟨ζ, E⟩ ≠ 0 for all ζ ∈ Z_{n}^{m}. If ⟨ζ, E⟩ < ∞ for some ζ_{1}, …, ζ_{m} ∈ ℤ^{∗}, then E fails (ℐ_{|ζ_{1}|+...+|ζ_{m}|}). Thus E enjoys (ℐ_{∞}) if and only if ⟨ζ, E⟩ = ∞ for all ζ_{1}, …, ζ_{m} ∈ ℤ^{∗}. (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. (iii) Suppose jF + s, kF + t ⊆ E for an infinite F, j ≠ k ∈ ℤ^{∗} and s, t ∈ ℤ. Then E fails (ℐ_{|j|+|k|}). (iv) Let E′ = {n_{k} + m_{k}} with {m_{k}} bounded. Then ⟨ζ, E⟩ = ∞ if and only if ⟨ζ, E′⟩ = ∞. Thus (ℐ_{∞}) is stable under bounded perturbations of E.

Proof. (i) Suppose ⟨ζ, E⟩ < ∞. Then there is an h ∈ ℤ such that there are sequences p_{1}^{l}, …, p_{m}^{l} ∈ {n_{k}}_{k≥l} with ∑ ζ_{i}p_{i}^{l} = h and {p_{1}^{l+1}, …, p_{m}^{l+1}} is disjoint from {p_{1}^{l}, …, p_{m}^{l}} for all l ≥ 1. As ∑ ζ_{i}p_{i}^{l} − ∑ ζ_{i}p_{i}^{l+1} = 0 for l ≥ 1, E fails (ℐ_{|ζ_{1}|+...+|ζ_{m}|}). (ii) Indeed, ⟨(1, −1), E⟩ = ∞. (iii) Put ζ = (j, −k). Then ⟨ζ, E⟩ < ∞.

3.2 Geometric sequences

Let G = {j^{k}}_{k≥0} with j ∈ ℤ \ {−1, 0, 1}. Then G, jG ⊆ G: so G fails (ℐ_{|j|+1}). In order to check (ℐ_{|j|}) for G, let us study more carefully the following Diophantine equation:

∑_{i=1}^{m} ζ_{i}j^{k_{i}} = 0 with ζ ∈ ℕ^{∗} × ℤ^{∗}^{m−1} & ∑_{i=1}^{m} |ζ_{i}| ≤ 2|j| & k_{1} < … < k_{m}. (7)

Suppose (7) holds. Then necessarily m ≥ 2 and ζ_{1} + ∑_{i=2}^{m} ζ_{i}j^{k_{i}−k_{1}} = 0. Hence j | ζ_{1} and ζ_{1} ≥ |j|. As ζ_{1} < 2|j|, ζ_{1} = |j|. Then sgn j + ∑_{i=2}^{m} ζ_{i}j^{k_{i}−k_{1}−1} = 0. Hence k_{2} = k_{1} + 1 and j | sgn j + ζ_{2}. As |ζ_{2}| ≤ |j|, ζ_{2} ∈ {− sgn j, j − sgn j}. If ζ_{2} = j − sgn j, then m = 3, k_{3} = k_{1} + 2 and ζ_{3} = −1. If ζ_{2} = − sgn j, then m = 2: otherwise, j | ζ_{3} as before and |ζ_{1}| + |ζ_{2}| + |ζ_{3}| > 2|j|. Thus (7) has exactly two solutions:

|j| · j^{k} + (− sgn j) · j^{k+1} = 0 & |j| · j^{k} + (j − sgn j) · j^{k+1} + (−1) · j^{k+2} = 0. (8)

If j is positive, this shows that G enjoys (ℐ_{j}): both solutions yield ∑ ζ_{i} ≠ 0. If j is negative, G enjoys (ℐ_{|j|−1}), but the second solution of (7) shows that G fails (ℐ_{|j|}).

3.3 Algebraic and transcendental numbers

An interesting feature of property (ℐ_{∞}) is that it distinguishes between algebraic and transcendental numbers. A similar fact has already been noticed by Murai [72, Prop. 26, Cor. 28].

Proposition 3.3.1 Let E = {n_{k}} ⊆ ℤ.

(i) If n_{k+1}/n_{k} → σ where σ > 1 is transcendental, then ⟨ζ, E⟩ = ∞ for any ζ_{1}, …, ζ_{m} ∈ ℤ^{∗}. Thus E enjoys (ℐ_{∞}). (ii) Write [x] for the integer part of a real x. Let n_{k} = [σ^{k}] with σ > 1 algebraic. Let P (x) = ζ_{0} +…+ζ_{d}x^{d} be the corresponding polynomial of minimal degree. Then ⟨ζ, E⟩ < ∞ and E fails (ℐ_{|ζ_{0}|+...+|ζ_{d}|}).

Note that part (ii) is very restrictive on the speed of convergence of n_{k+1}/n_{k} to σ: even if we take into account Proposition 3.1.1(iv), it requires that

|n_{k+1}/n_{k} − σ| ≼ σ^{−k}.

Proof. (i) Suppose on the contrary that we have ζ and sequences p_{1}^{l} < … < p_{m}^{l} in E that tend to infinity such that ζ_{1}p_{1}^{l} + … + ζ_{m}p_{m}^{l} = 0. As the sequences {p_{i}^{l}/p_{m}^{l}}_{l} (1 ≤ i ≤ m) are bounded, we may assume they are converging — and by hypothesis, they converge either to 0, say for i < j, or to σ^{−d_{i}} for d_{i} ∈ ℕ and i ≥ j. But then ζ_{j}σ^{−d_{j}} + … + ζ_{m}σ^{−d_{m}} = 0 and σ is algebraic. (ii) Apply Proposition 3.1.1(i) with ζ:

|ζ_{0}[σ^{k}] + … + ζ_{d}[σ^{k+d}]| = |ζ_{0}([σ^{k}] − σ^{k}) + … + ζ_{d}([σ^{k+d}] − σ^{k+d})| ≤ ∑ |ζ_{i}|.

3.4 Polynomial sequences

Let us first give some numerical evidence for the classical case of sets of dth 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.

{k^{d}} fails by counterexample
d=2 (ℐ_{2}) 7^{2} + 1^{2} = 2 · 5^{2} (or 18^{2} + 1^{2} = 15^{2} + 10^{2} [16, book II, problem 9])
d=3 (ℐ_{2}) 12^{3} + 1^{3} = 10^{3} + 9^{3} [11, due to Frénicle]
d=4 (ℐ_{2}) 158^{4} + 59^{4} = 134^{4} + 133^{4} (or 12231^{4} + 2903^{4} = 10381^{4} + 10203^{4} [25])
d=5 (ℐ_{3}) 67^{5} + 28^{5} + 24^{5} = 62^{5} + 54^{5} + 3^{5} (another first in [70])
d=6 (ℐ_{3}) 23^{6} + 15^{6} + 10^{6} = 22^{6} + 19^{6} + 3^{6} [82]
d=7 (ℐ_{4}) 149^{7} + 123^{7} + 14^{7} + 10^{7} = 146^{7} + 129^{7} + 90^{7} + 15^{7} [18]
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 [19])
d=9 (ℐ_{6}) 23^{9} + 18^{9} + 14^{9} + 2 · 13^{9} + 1^{9} = 22^{9} + 21^{9} + 15^{9} + 10^{9} + 9^{9} + 5^{9} [56]
d=10 (ℐ_{7}) 38^{10} + 33^{10} + 2 · 26^{10} + 15^{10} + 8^{10} + 1^{10} =

36^{10} + 35^{10} + 32^{10} + 29^{10} + 24^{10} + 23^{10} + 22^{10} (another first in [70])

Table 3.4.1

Note that a positive answer to Euler’s conjecture — for k ≥ 5 a^{k} + b^{k} = c^{k} + d^{k} has only trivial solutions in integers — would imply that the set of kth powers has (ℐ_{2}). This conjecture has been neither proved nor disproved for any value of k ≥ 5 (see [91] and [19]). Now let E = {n_{k}} ⊆ ℤ be a set of polynomial growth: |n_{k}| ≼ k^{d} for some d ≥ 1. Then E ∩ [−n, n] ≽ n^{1/d} and by [86, Th. 3.6], E fails the Λ(p) property for p > 2d and E fails 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. Recall that

Δ^{j}P (k) = ∑_{i=0}^{j} ({j \atop i})(−1)^{i}P (k − i) , ∑_{i=0}^{j} ({j \atop i})(−1)^{i} = 0 , ∑_{i=0}^{j} ({j \atop i}) = 2^{j}. (9)

As Δ^{d+1}P (k) = 0, this makes E fail (ℐ_{2^{d}}), which is coarse.

Conclusion By Theorem 2.4.2, property (ℐ_{n}) yields directly (umbs) in the spaces L^{2p}(𝕋), p ≤ n integer. But we do not know whether (ℐ_{∞}) ensures (umbs) in spaces L^{p}(𝕋), p not an even integer.

4 Metric unconditional approximation property

As we investigate simultaneously real and complex (umap), it is convenient to introduce a subgroup 𝕊 of 𝕋 corresponding to each case. Thus, if 𝕊 = 𝔻 = {−1, 1}, then the following applies to real (umap). If 𝕊 = 𝕋 = {ϵ ∈ ℂ : |ϵ| = 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) ⇔ (iv) in Theorem 4.3.1 and then pass on to Section 6.

4.1 Definition

We start with defining the metric unconditional approximation property ((umap) for short). Recall that ΔT_{k} = T_{k} − T_{k−1} (where T_{0} = 0).

Definition 4.1.1 Let X be a separable Banach space.

(i) A sequence {T_{k}} of operators on X is an approximating sequence (a.s. for short) if each T_{k} has finite rank and ∥T_{k}x − x∥ → 0 for every x ∈ X. If X admits an a.s., it has the bounded approximation property. An a.s. of commuting projections is a finite-dimensional decomposition ((fdd) for short). (ii) [27] X has the unconditional approximation property (uap) if there are an a.s. {T_{k}} and a constant C such that

∥∑_{k=1}^{n} ϵ_{k}ΔT_{k}∥ ≤ C for all n and ϵ_{k} ∈ 𝕊.  (10)

The (uap) constant is the least such C. (iii) [12, §3] X has the metric unconditional approximation property (umap) if it has (uap) with constant 1 + ε for any ε > 0.

Property (ii) is the approximation property which most appropriately generalizes the unconditional basis property. It has first been introduced by Pełczyński and Wojtaszczyk [75]. They showed that it holds if and only if X is a complemented subspace of a space with an unconditional (fdd). By [60, Th. 1.g.5], this implies that X is subspace of a space with an unconditional basis. Thus, neither L^{1}([0, 1]) nor 𝒞([0, 1]) share (uap). Property (iii) has been introduced by Casazza and Kalton as an extreme form of metric approximation. It has been studied in [12, §3], [33, §8,9], [32] and [31, §IV]. There is a simple and very useful criterion for (umap):

Proposition 4.1.2 ([12, Th. 3.8] and [33, Lemma 8.1]) Let X be a separable Banach space. X has (umap) if and only if there is an a.s. {T_{k}} such that

sup_{ϵ∈𝕊} ∥(Id − T_{k}) + ϵT_{k}∥ ⟶_{k→∞} 1.  (11)

If (11) holds, we say that {T_{k}} realizes (umap). A careful reading of the above mentioned proof also gives the following results for a.s. that satisfy T_{n+1}T_{n} = T_{n}.

Proposition 4.1.3 Let X be a separable Banach space.

(i) Let {T_{k}} be an a.s. 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 ≥ 1 and ϵ ∈ 𝕊

∥Id − (1 + ϵ)T′_{k}∥ = 1.

(ii) X has metric unconditional (fdd) if and only if there is an (fdd) {T_{k}} such that (11) holds.

4.2 Characterization of (umap). Block unconditionality

We want to characterize (umap) in an even simpler way than Proposition 4.1.2. Relation (11) and the method of [48, Th. 4.2], suggest considering some unconditionality condition between a certain “break” and a certain “tail” of X. We propose two such notions.

Definition 4.2.1 Let X be a separable Banach space.

(i) Let τ be a vector space topology on X. Then X has the property (u(τ)) of τunconditionality if for all u ∈ X and norm bounded sequences {v_{j}} ⊆ X such that

v_{j} →^{τ} 0

osc_{ϵ∈𝕊} ∥ϵu + v_{j}∥ → 0.  (12)

(ii) Let {T_{k}} be a commuting a.s. X has the property (u(T_{k})) of commuting block unconditionality if for all ε > 0 and n ≥ 1 we may choose m ≥ n such that for all x ∈ T_{n}B_{X} and y ∈ (Id − T_{m})B_{X}

osc_{ϵ∈𝕊} ∥ϵx + y∥ ≤ ε.  (13)

Thus, given a commuting a.s. {T_{k}}, T_{n}X is the “break” and (Id − T_{m})X the “tail” of X. We have

Lemma 4.2.2 Let X be a separable Banach space and {T_{k}} a commuting a.s. for X. The following are equivalent.

(i) X enjoys (u(τ)) for some vector space topology τ such that T_{n}x →^{τ} x uniformly for x ∈ B_{X}; (ii) X enjoys (u(T_{k})).

Proof. (i) ⇒ (ii). Suppose that (ii) fails: there are n ≥ 1 and ε > 0 such that for each m ≥ n, there are x_{m} ∈ T_{n}B_{X} and y_{m} ∈ (Id − T_{m})B_{X} such that

osc_{ϵ∈𝕊} ∥ϵx_{m} + y_{m}∥ > ε.

As T_{n}B_{X} is compact, we may suppose by extracting a convergent subsequence that x_{m} = x. Let τ be as in (i): then y_{m} →^{τ} 0 and (u(τ)) must fail. (ii) ⇒ (i). Let us define a vector space topology τ by

x_{n} →^{τ} 0 ⟺ ∀k ∥T_{k}x_{n}∥ → 0.

Then T_{n}x →^{τ} x uniformly on B_{X}. Indeed, T_{k}(T_{n}x − x) = (T_{n} − Id)T_{k}x and T_{n} − Id converges uniformly to 0 on T_{k}B_{X} which is norm compact. Let us check (u(τ)). Let u ∈ B_{X} and {v_{j}} ⊆ B_{X} be such that v_{j} →^{τ} 0. Let ε > 0. There is n ≥ 1 such that ∥T_{n}u − u∥ ≤ ε. Choose m such that (13) holds for x ∈ T_{n}B_{X} and y ∈ (Id − T_{m})B_{X}. Then choose k ≥ 1 such that ∥T_{m}v_{j}∥ ≤ ε for j ≥ k. We have, for any ϵ ∈ 𝕊,

∥ϵu + v_{j}∥ ≤ ∥ϵT_{n}u + (Id − T_{m})v_{j}∥ + ∥T_{n}u − u∥ + ∥T_{m}v_{j}∥
≤ ∥T_{n}u + (Id − T_{m})v_{j}∥ + 3ε ≤ ∥u + v_{j}∥ + 5ε.

Thus we have (12). In order to obtain (umap) from block independence, we shall have to construct unconditional skipped blocking decompositions.

Definition 4.2.3 Let X be a separable Banach space. X admits unconditional skipped blocking decompositions if for each ε > 0, there is an unconditional a.s. {S_{k}} such that for all 0 ≤ a_{1} < b_{1} < a_{2} < b_{2} < … and x_{k} ∈ (S_{b_{k}} − S_{a_{k}})X

sup_{ϵ_{k}∈𝕊}∥∑ ϵ_{k}x_{k}∥ ≤ (1 + ε)∥∑ x_{k}∥.

4.3 Main theorem: convex combinations of multipliers

We have

Theorem 4.3.1 Consider the following properties for a separable Banach space X.

(i) There are an unconditional commuting a.s. {T_{k}} and a vector space topology τ such that X enjoys (u(τ)) and T_{k}x →^{τ} x uniformly for x ∈ B_{X}; (ii) X enjoys (u(T_{k})) for an unconditional commuting a.s. {T_{k}}; (iii) X admits unconditional skipped blocking decompositions; (iv) X has (umap). Then (iv) ⇒ (i) ⇔ (ii) ⇒ (iii). If X has finite cotype, then (iii) ⇒ (iv).

Proof. (i) ⇔ (ii) holds by Lemma 4.2.2. (iv) ⇒ (ii). By Godefroy–Kalton’s [31, Th. IV.1], there is in fact an a.s. {T_{k}} that satisfies (11) such that T_{k}T_{l} = T_{min(k,l)} if k ≠ l. Let C be a uniform bound for ∥T_{k}∥. Let ε > 0 and n ≥ 1. There is m ≥ n + 2 such that

sup_{ϵ∈𝕊} ∥ϵT_{m−1} + (Id − T_{m−1})∥ ≤ 1 + ε/2C.

Let x ∈ T_{n}B_{X} and y ∈ (Id − T_{m})B_{X}. As x − T_{m−1}x = 0 and T_{m−1}y = 0,

ϵx + y = ϵT_{m−1}(x + y) + (Id − T_{m−1})(x + y),

and, for all ϵ ∈ 𝕊,

∥ϵx + y∥ ≤ (1 + ε/2C)∥x + y∥ ≤ ∥x + y∥ + ε.

(ii) ⇒ (iii). By a perturbation [90, proof of Lemma III.9.2], we may suppose that T_{k}T_{l} = T_{min(k,l)} if k ≠ l. Let ε > 0 and choose a sequence of η_{j} > 0 such that 1 + ε_{j} = ∏_{i≤j}(1 + η_{i}) < 1 + ε 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

sup_{ϵ∈𝕊} ∥x + ϵy∥ ≤ (1 + η_{j})∥x + y∥  (14)

for x ∈ (Id − S_{j})X and y ∈ S_{j−1}X. Let us show that it is an unconditional skipped blocking decomposition: we shall prove by induction that

(H_{j})  \{ \begin{matrix}sup_{ϵ_{i}∈𝕊}∥x + ∑_{i=1}^{n} ϵ_{i}x_{i}∥ ≤ (1 + ε_{j})∥x + ∑_{i=1}^{n} x_{i}∥ for x ∈ (Id − S_{j})X \\ and x_{i} ∈ (S_{b_{i}} − S_{a_{i}})X (0 ≤ a_{1} < b_{1} < … < a_{n} < b_{n} ≤ j − 1). \end{matrix}

(H_{1}) trivially holds.

Assume (H_{i}) holds for i < j. Let x and x_{i} as in (H_{j}). Let ϵ_{i} ∈ 𝕊. Then

∥x + ∑_{i=1}^{n} ϵ_{i}x_{i}∥ ≤ (1 + η_{j})∥x + \overline{ϵ_{n}} ∑_{i=1}^{n} ϵ_{i}x_{i}∥ = (1 + η_{j})∥x + x_{n} + ∑_{i=1}^{n−1} \overline{ϵ_{n}}ϵ_{i}x_{i}∥

by (14). Note that x + x_{n} ∈ (Id − S_{a_{n}})X: an application of (H_{a_{n}}) yields (H_{j}). (iii) ⇒ (iv). Let ε > 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} for 1 ≤ j ≤ n and i ≥ 0.

The jth skipped blocks are

U_{j} = Id − ∑_{i} V_{i,j} = ∑_{i} ΔS_{in+j};

then ∑_{j=1}^{n} U_{j} = Id. Let

R_{i} = \frac{1}{n − 1} ∑_{j=1}^{n} V_{i,j};

then R_{i} has finite rank and

R_{0} + R_{1} + … = (nId − Id)/(n − 1) = Id.

Thus W_{j} = ∑_{i≤j} R_{i} defines an a.s. We may bound its (uap) constant. First, since {S_{k}} is a skipped blocking decomposition,

∀x ∈ B_{X} sup_{ϵ_{i}∈𝕊}∥∑ ϵ_{i}R_{i}x∥ ≤ \frac{1}{n − 1} ∑_{j=1}^{n} sup_{ϵ_{i}∈𝕊}∥∑_{i} ϵ_{i}V_{i,j}x∥
≤ \frac{1 + ε}{n − 1} ∑_{j=1}^{n} ∥x − U_{j}x∥
≤ \frac{1 + ε}{n − 1}(n + ∑_{j=1}^{n} ∥U_{j}x∥).

Let us bound ∑_{1}^{n} ∥U_{j}x∥. Let q < ∞ be the cotype of X and C_{c} its cotype constant. Then by Hölder’s inequality we have for all x ∈ B_{X}

∑ ∥U_{j}x∥ ≤ n^{1−1/q}(∑ ∥U_{j}x∥^{q})^{1/q}
≤ n^{1−1/q}C_{c} · average_{±}∥∑ ±U_{j}x∥ ≤ n^{1−1/q}C_{c}C_{u}.  (15)

Thus the (uap) constant of {W_{j}} is at most (1 + ε)(n + C_{c}C_{u}n^{1−1/q})/(n − 1). As ε is arbitrarily little and n arbitrarily large, X has (umap).

Remark 4.3.2 How does Theorem 4.3.1 look in the special cases where τ is the weak or the weak^{∗} topology ? They correspond to the classical cases where the a.s. is shrinking vs. boundedly complete.

We may remove the cotype assumption in Theorem 4.3.1 (iii) ⇒ (iv) if the space has the properties of commuting ℓ_{1}-(ap) or ℓ_{q}-(fdd) for q < ∞, which will be introduced in Section 5:

Theorem 4.3.3 Consider the following properties for a separable Banach space X.

(i) There are a commuting ℓ_{1}-a.s. or an ℓ_{q}-(fdd) {T_{k}}, q < ∞, and a vector space topology τ such that X enjoys (u(τ)) and T_{k}x →^{τ} x uniformly for x ∈ B_{X}; (ii) X enjoys (u(T_{k})) for a commmuting ℓ_{1}-a.s. or an ℓ_{q}-(fdd) {T_{k}}, q < ∞; (iii) X admits unconditional skipped blocking decompositions and one may in fact take an ℓ_{1}-a.s. or an ℓ_{q}-(fdd) {T_{k}}, q < ∞, in its definition 4.2.3; (iv) X has (umap). Then (i) ⇔ (ii) ⇒ (iii) ⇒ (iv).

Proof. Part (i) ⇔ (ii) ⇒ (iii) goes as before. To prove (iii) ⇒ (iv), note that in the proof of Theorem 4.3.1 (iii) ⇒ (iv), one may replace the estimate in (15) by

∀x ∈ B_{X} ∑ ∥U_{j}x∥ ≤ n^{1−1/q}(∑ ∥U_{j}x∥^{q})^{1/q} ≤ n^{1−1/q}C_{ℓ},

where C_{ℓ} is the ℓ_{1}-(ap) or the ℓ_{q}-(fdd) constant.

5 The p-additive approximation property ℓ_{p}-(ap)

5.1 Definition

Definition 5.1.1 Let X be a separable Banach space.

(i) X has the p-additive approximation property ℓ_{p}-(ap) if there are an a.s. {T_{k}} and a constant C such that

C^{−1}∥x∥ ≤ (∑ ∥ΔT_{k}x∥^{p})^{1/p} ≤ C∥x∥  (16)

for all x ∈ X. The ℓ_{p}-(ap) constant is the least such C. (ii) X has the metric p-additive approximation property ℓ_{p}-(map) if it has ℓ_{p}-(ap) with constant 1 + ε for any ε > 0.

Note that ℓ_{p}-(ap) implies (uap) and ℓ_{p}-(map) implies (umap). Note also that in (16), the left inequality is trivial with C = 1 if p = 1; the right inequality is always achieved for some C if p = ∞. Property (ii) is implicit in Kalton–Werner’s [48] investigation of subspaces of L^{p} that are almost isometric to subspaces of ℓ_{p}: see Section 5.4. The proof of Proposition 4.1.2 can be adapted to yield

Proposition 5.1.2 Let X be a separable Banach space.

(i) If there is an a.s. {T_{k}} such that

(∥x − T_{k}x∥^{p} + ∥T_{k}x∥^{p})^{1/p} ⟶_{k→∞} 1  (17)

uniformly on the unit sphere, then X has ℓ_{p}-(map). The converse holds if p = 1. (ii) X has a metric ℓ_{p}-(fdd) if and only if there is an (fdd) {T_{k}} such that (17) holds.

5. the p-additive approximation property ℓ_{p}-(ap) 43

We shall say that {T_{k}} realizes ℓ_{p}-(map) if it satisfies (17).

Proof. Let {T_{k}} be an a.s. that satisfies (17) and ε > 0. By a perturbation [42, Lemma 2.4], we may suppose that T_{k+1}T_{k} = T_{k}. Choose a sequence of η_{j} > 0 such that 1 + ε_{k} = ∏_{j≤k}(1 + η_{j}) ≤ 1 + ε for each k. We may assume by taking a subsequence of the T_{k}’s that for all k and x ∈ X,

(1 + η_{k})^{−1}∥x∥ ≤ (∥x − T_{k}x∥^{p} + ∥T_{k}x∥^{p})^{1/p} ≤ (1 + η_{k})∥x∥.  (18)

We then prove by induction the hypothesis (H_{k})

∀x ∈ X (1 + ε_{k})^{−1}∥x∥ ≤ (∥x − T_{k}x∥^{p} + ∑_{j=1}^{k} ∥ΔT_{j}x∥^{p})^{1/p} ≤ (1 + ε_{k})∥x∥.

(H_{1}) is true. Suppose (H_{k−1}) is true. Let x ∈ X. Note that

x − T_{k}x = (Id − T_{k})(x − T_{k−1}x) , ΔT_{k}x = T_{k}(x − T_{k−1}x).

By (18), we get

(∥x − T_{k}x∥^{p} + ∥ΔT_{k}x∥^{p})^{1/p} ≤ (1 + η_{k})∥x − T_{k−1}x∥.

Hence, by (H_{k−1}),

(∥x − T_{k}x∥^{p} + ∑_{j=1}^{k} ∥ΔT_{j}x∥^{p})^{1/p} ≤

≤ (1 + η_{k})(∥x − T_{k−1}x∥^{p} + ∑_{j=1}^{k−1} ∥ΔT_{j}x∥^{p})^{1/p} ≤ (1 + ε_{k})∥x∥.

We obtain the lower bound in the same way. Thus the induction is complete. Hence {T_{k}} realizes ℓ_{p}-(ap) with constant 1 + ε. As ε is arbitrary, X has ℓ_{p}-(map). If X has ℓ_{1}-(map), then for each ε > 0, there is a sequence {S_{k}} such that

∥x∥ ≤ ∥x − S_{k}x∥ + ∥S_{k}x∥ ≤ ∑ ∥ΔS_{k}x∥ ≤ (1 + ε)∥x∥

for all x ∈ X. By a diagonal argument, this gives an a.s. {T_{k}} satisfying (17). (iii) If X has a metric ℓ_{p}-(fdd), then for each ε > 0 there is a (fdd) {T_{k}} such that (16) holds with C = 1 + ε. Then, for all k ≥ 1,

(1 − ε)∥T_{k}x∥ ≤ (∑_{j=1}^{k} ∥ΔT_{j}x∥^{p})^{1/p} ≤ (1 + ε)∥T_{k}x∥

(1 − ε)∥x − T_{k}x∥ ≤ ( ∑_{j=k+1}^{∞} ∥ΔT_{j}x∥^{p})^{1/p} ≤ (1 + ε)∥x − T_{k}x∥.

Thus

(1 − ε)/(1 + ε)∥x∥ ≤ (∥x − T_{k}x∥^{p} + ∥T_{k}x∥^{p})^{1/p} ≤ (1 + ε)/(1 − ε)∥x∥.

By a diagonal argument, this gives an (fdd) {T_{k}} satisfying (17).

Question 5.1.3 What about the converse in Proposition 5.1.2(i) for p > 1 ?

5.2 Some consequences of ℓ_{p}-(ap)

We start with the simple

Proposition 5.2.1 Let X be a separable Banach space.

(i) If X has ℓ_{p}-(ap) with constant C, then X is C-isomorphic to a subspace of an ℓ_{p}-sum of finite dimensional subspaces of X. (ii) If furthermore X is a subspace of L^{q}, then X is (C +ε)-isomorphic to a subspace of (⨁ ℓ_{q}^{n})_{p} for any given ε > 0. (iii) In particular, if a subspace of L^{p} has ℓ_{p}-(ap) with constant C, then it is (C +ε)- isomorphic to a subspace of ℓ_{p} for any given ε > 0. If a subspace of L^{p} has ℓ_{p}-(map), then it is almost isometric to subspaces of ℓ_{p}.

Proof. (i) Indeed, Φ: X ↪ (⨁ im ΔT_{i})_{p}, x ↦ {ΔT_{i}x}_{i≥1} is an embedding: for all x ∈ X

C^{−1}∥x∥_{X} ≤ ∥Φx∥ = (∑ ∥ΔT_{i}x∥_{X}^{p})^{1/p} ≤ C∥x∥_{X}.

(ii & iii) Recall that, given ε > 0, a finite dimensional subspace of L^{q} is (1 + ε)- isomorphic to a subspace of ℓ_{q}^{n} for some n ≥ 1.

We have in particular (see [41, §VIII, Def. 7] for the definition of Hilbert sets)

Corollary 5.2.2 Let E ⊆ ℤ be infinite.

(i) No L_{E}^{q}(𝕋) (1 ≤ q < ∞) has ℓ_{p}-(ap) for p ≠ 2. (ii) No 𝒞_{E}(𝕋) has ℓ_{q}-(ap) for q ≠ 1. If E is a Hilbert set, then 𝒞_{E}(𝕋) fails ℓ_{1}-(ap).

Proof. This is a consequence of Proposition 5.2.1(i): every infinite E contains a Sidon set and thus a Λ(2 ∨ p) set. So L_{E}^{p}(𝕋) contains ℓ_{2}. Also, if E is a Hilbert set, then 𝒞_{E}(𝕋) contains c_{0} by [57, Th. 2].

However, there is a Hilbert set E such that 𝒞_{E}(𝕋) has complex (umap): see [58, Th. 10]. The class of sets E such that 𝒞_{E}(𝕋) has ℓ_{1}-(ap) contains the Sidon sets and Blei’s sup-norm-partitioned sets [7].

5.3 Characterization of ℓ_{p}-(map)

Recall [48, Def. 4.1]:

Definition 5.3.1 Let X be a separable Banach space.

(i) Let τ be a vector space topology on X. X enjoys property (m_{p}(τ)) if for all x ∈ X and norm bounded sequences {y_{j}} such that y_{j} →^{τ} 0

|∥x + y_{j}∥ − (∥x∥^{p} + ∥y_{j}∥^{p})^{1/p}| → 0.

(ii) X enjoys the property (m_{p}(T_{k})) for a commuting a.s. {T_{k}} if for all ε > 0 and n ≥ 1 we may choose m ≥ n such that for all x ∈ B_{X}

|∥T_{n}x + (Id − T_{m})x∥ − (∥T_{n}x∥^{p} + ∥(Id − T_{m})x∥^{p})^{1/p}| ≤ ε.

Then [48, Th. 4.2] may be read as follows

Theorem 5.3.2 Let 1 ≤ p < ∞ and consider the following properties for a separable Banach space X.

(i) There are an unconditional commuting a.s. {T_{k}} and a vector space topology τ such that X enjoys (m_{p}(τ)) and T_{k}x →^{τ} x uniformly for x ∈ B_{X}; (ii) X enjoys the property (m_{p}(T_{k})) for an unconditional commuting a.s. {T_{k}}. (iii) X has ℓ_{p}-(map). Then (i) ⇔ (ii). If X has finite cotype, then (ii) ⇒ (iii).

As for Theorem 4.3.1, we may remove the cotype assumption if X has commuting ℓ_{1}-(ap) or ℓ_{p}-(fdd), p < ∞:

Theorem 5.3.3 Let 1 ≤ p < ∞. Consider the following properties for a separable Banach space X.

(i) There are an ℓ_{p}-(fdd) (or just a commuting ℓ_{1}-a.s. in the case p = 1) {T_{k}} and a vector space topology τ such that X enjoys (m_{p}(τ)) and T_{k}x →^{τ} x uniformly for x ∈ B_{X}; (ii) X enjoys (m_{p}(T_{k})) for an ℓ_{p}-(fdd) (or just a commuting ℓ_{1}-a.s. in the case p = 1) {T_{k}}. (iii) X has ℓ_{p}-(map). Then (i) ⇔ (ii) ⇒ (iii).

5.4 Subspaces of L^{p} with ℓ_{p}-(map)

Although no translation invariant subspace of L^{p}(𝕋) has ℓ_{p}-(ap) for p ≠ 2, Proposition 5.2.1 (iii) is not void. By the work of Godefroy, Kalton, Li and Werner [48, 32], we get examples of subspaces of L^{p} with ℓ_{p}-(map) 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 property when, given δ ∈ ]0, 2] and ε > 0, any normalized δ-separated sequence in X contains a subsequence that is (2/δ + ε)-equivalent to the unit vector basis of ℓ_{1} (see [85]). In particular, a gliding hump argument shows that any subspace of ℓ_{1} shares this property. By Proposition 5.2.1(iii), a space X with ℓ_{1}-(map) also does. Now recall the main theorem of [32]:

Theorem Let X be a subspace of 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 + ε)-isomorphic to a w^{∗}-closed subspace X_{ε} of ℓ_{1} for any ε > 0.

We may then add to these three the fourth equivalent property

(iv) X has ℓ_{1}-(map).

Proof. We just showed that (ii) holds when X has ℓ_{1}-(map). Now suppose we have (iii) and let ε > 0. Thus there is a quotient Z of c_{0} such that Z^{∗} has the approximation property and Z^{∗} is (1 + ε)-isomorphic to X. Let us show that any such Z^{∗} has ℓ_{1}-(map). Z has beforehand the metric approximation property, with say {R_{n}}, because Z^{∗} has it as a dual separable space. By [34, Th. 2.2], {R_{n}^{∗}} is a metric a.s. 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 a.s. in ℓ_{1}. Thus

∥P_{n}^{∗}Q^{∗}x^{∗} − Q^{∗}R_{n}^{∗}x^{∗}∥_{ℓ_{1}} → 0 for any x^{∗} ∈ Z^{∗}.

By Lebesgue’s dominated convergence theorem (see [46, Th. 1]), QP_{n} − R_{n}Q → 0 weakly in the space 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_{n}Q∥_{ℒ(c_{0},Z)} → 0. Thus

∥C_{n}^{∗}Q^{∗} − Q^{∗}D_{n}^{∗}∥_{ℒ(Z^{∗},ℓ_{1})} → 0.  (19)

Furthermore C_{n}^{∗} : ℓ_{1} → ℓ_{1} has the form C_{n}^{∗}(x_{1}, x_{2}, …) = (t_{1}x_{1}, t_{2}x_{2}, …) with 0 ≤ t_{i} ≤ 1. Therefore, defining Q^{∗}a = (a_{1}, a_{2}, …),

∥C_{n}^{∗}Q^{∗}a∥_{1} + ∥Q^{∗}a − C_{n}^{∗}Q^{∗}a∥_{1} =
= ∥(t_{1}a_{1}, t_{2}a_{2}, …)∥_{1} + ∥((1 − t_{1})a_{1}, (1 − t_{2})a_{2}, …)∥_{1}
= ∑(|t_{i}| + |1 − t_{i}|)|a_{i}| = ∑ |a_{i}| = ∥Q^{∗}a∥_{1}.  (20)

As {D_{n}^{∗}} is still an a.s. for Z^{∗}, {D_{n}^{∗}} realizes ℓ_{1}-(map) in Z^{∗} by (20), (19) and Proposition 5.1.2(i). Thus X has ℓ_{1}-(ap) with constant 1 + 2ε. As ε is arbitrary, X has ℓ_{1}-(map).

For 1 < p < ∞, we have similarly by [48, Th. 4.2] Proposition 5.4.1 Let 1 < p < ∞ and X be a subspace of L^{p} with the approximation property. The following are equivalent:

(i) X is (1 + ε)-isomorphic to a subspace X_{ε} of ℓ_{p} for any ε > 0. (ii) X has ℓ_{p}-(map). Proof. (ii) ⇒ (i) is in Proposition 5.2.1. For (i) ⇒ (ii), it suffices to prove that any subspace Z of ℓ_{p} with the approximation property has ℓ_{p}-(map). As Z is reflexive, Z admits a commuting shrinking a.s. {R_{n}}. Let i be the injection of Z into ℓ_{p}. Let {P_{n}} be the sequence of projections associated to the natural basis of ℓ_{p}. It is also an a.s. for ℓ_{p′}. Thus

∥i^{∗}P_{n}^{∗}x^{∗} − R_{n}^{∗}i^{∗}x^{∗}∥_{Z^{∗}} → 0 for any x^{∗} ∈ ℓ_{p′}.

As before, there are convex combinations {C_{n}} of {P_{n}} and {D_{n}} of {R_{n}} such that ∥C_{n}i − iD_{n}∥ → 0. The convex combinations are finite and may be chosen not to overlap, so that for each n ≥ 1 there is m > n such that

∥C_{n}x + (Id − C_{m})x∥ = (∥C_{n}x∥^{p} + ∥(Id − C_{m})x∥^{p})^{1/p}

for x ∈ ℓ_{p}. Thus Z satisfies the property (m_{p}(D_{n})). Following the lines of [27, Lemma 1], we observe that {D_{n}} is a commuting unconditional a.s. since {P_{n}} is. By Theorem 5.3.2, Z has ℓ_{p}-(map).

6 (uap) and (umap) in translation invariant spaces

Recall that 𝕊 is a subgroup of 𝕋. If 𝕊 = 𝔻 = {−1, 1}, the following applies to real (umap). If 𝕊 = 𝕋 = {ϵ ∈ ℂ : |ϵ| = 1}, it applies to complex (umap).

6.1 General properties. Isomorphic case

L^{p}(𝕋) spaces (1 < p < ∞) are known to have an unconditional basis; furthermore, they have an unconditional (fdd) in translation invariant subspaces L_{I_{k}}^{p}(𝕋): this is a corollary of Littlewood–Paley theory [61]. One may choose I_{0} = {0} and I_{k} = ]−2^{k}, −2^{k−1}] ∪ [2^{k−1}, 2^{k}[. Thus any L_{E}^{p}(𝕋) (1 < p < ∞) has an unconditional (fdd) in translation invariant subspaces L_{E∩I_{k}}^{p}(𝕋). The spaces L^{1}(𝕋) and 𝒞(𝕋), however, do not even have (uap).

Proposition 6.1.1 (see [58, Lemma 5, Cor. 6, Th. 7]) Let E ⊆ ℤ and X be a homogeneous Banach space on 𝕋.

(i) If X_{E} has (umap) (vs. (uap), ℓ_{1}-(ap) or ℓ_{1}-(map)), then some a.s. of multipliers realizes it. (ii) Let F ⊆ E. If X_{E} has (umap) (vs. (uap), ℓ_{1}-(ap) or ℓ_{1}-(map)), then so does X_{F}. (iii) If 𝒞_{E}(𝕋) has (umap) (vs. (uap)), then so does X_{E}.

Note the important property that a.s. of multipliers commute and commute with one another. Whereas (uap) is always satisfied for L_{E}^{p}(𝕋) (1 < p < ∞), we have the following generalization of [58, remark after Th. 7, Prop. 9] for the spaces L_{E}^{1}(𝕋) and 𝒞_{E}(𝕋). By the method of [31],

Lemma 6.1.2 If X has (uap) with a commuting a.s. and X ⊉ c_{0}, then X is a dual space.

Proof. Suppose {T_{n}} is a commuting a.s. such that (10) holds. As X ⊉ c_{0}, P x^{∗∗} = lim T_{n}^{∗∗}x^{∗∗} is well defined for each x^{∗∗} ∈ X^{∗∗}. As {T_{n}} is an a.s., P is a projection onto X. Let us show that ker P is w^{∗}-closed. Indeed, if x^{∗∗} ∈ ker P, then

∥T_{n}^{∗∗}x^{∗∗}∥ = lim_{m} ∥T_{m}T_{n}^{∗∗}x^{∗∗}∥ = lim_{m} ∥T_{n}T_{m}^{∗∗}x^{∗∗}∥ = 0

and T_{n}^{∗∗}x^{∗∗} = 0. Thus

ker P = ⋂_{n} ker T_{n}^{∗∗}.

Let M = (ker P)_{⊥}. Then M^{∗} = X.

Corollary 6.1.3 Let E ⊆ ℤ.

(i) If L_{E}^{1}(𝕋) has (uap), then E is a Riesz set. (ii) If 𝒞_{E}(𝕋) has (uap) and 𝒞_{E}(𝕋) ⊉ c_{0}, then E is a Rosenthal set.

Proof. In both cases, Lemma 6.1.2 shows that the two spaces are separable dual spaces and thus have the Radon–Nikodym property. We may now apply Lust-Piquard’s characterization [63].

There are Riesz sets E such that L_{E}^{1}(𝕋) fails (uap): indeed, the family of Riesz sets is coanalytic [95] while the second condition is in fact analytic. There are Rosenthal sets that cannot be sup-norm-partitioned [7]. The converse of Proposition 6.1.1(iii) does not hold: L_{E}^{1}(𝕋) may have (uap) while 𝒞_{E}(𝕋) fails this property. We have

Proposition 6.1.4 Let E ⊆ ℤ.

(i) The Hardy space H^{1}(𝕋) = L_{ℕ}^{1}(𝕋) has (uap). (ii) The disc algebra A(𝕋) = 𝒞_{ℕ}(𝕋) fails (uap). More generally, if ℤ \ E is a Riesz set, then 𝒞_{E}(𝕋) fails (uap).

Proof. (i) Indeed, H^{1}(𝕋) has an unconditional basis [66]. Note that the first unconditional a.s. for H^{1}(𝕋) appears in [67, §II, introduction] with the help of Stein’s [92, 93] multiplier theorem (see also [99]). (ii) Let Δ ⊂ 𝕋 be the Cantor set. By Bishop’s improvement [6] of Rudin–Carleson’s interpolation theorem, every function in 𝒞(Δ) extends to a function in 𝒞_{E}(𝕋) if ℤ\E is a Riesz set. By [74, main theorem], this implies that 𝒞(Δ) embeds in 𝒞_{E}(𝕋). Then 𝒞_{E}(𝕋) cannot have (uap); otherwise 𝒞(Δ) would embed in a space with an unconditional basis, which is false.

Remark 6.1.5 Recent studies of the Daugavet Property by Kadets and Werner generalize Proposition 6.1.4(ii). This property of a Banach space X states that for every finite rank operator T on X ∥Id + T ∥ = 1 + ∥T ∥. By [43, Th. 2.1], such an X cannot have (uap). Further, by [97, Th. 3.7], 𝒞_{E}(𝕋) has the Daugavet Property if ℤ \ E is a so-called semi-Riesz set, that is if all measures with Fourier spectrum in ℤ \ E are diffuse.

Question 6.1.6 Is there some characterization of sets E ⊆ ℤ such that 𝒞_{E}(𝕋) has (uap) ? Only a few classes of such sets are known: Sidon sets and sup-norm-partitioned sets, for which 𝒞_{E}(𝕋) even has ℓ_{1}-(ap); certain Hilbert sets. We conjecture that 𝒞_{E}(𝕋) fails (uap) if E contains an infinite parallelepiped.

6.2 Characterization of (umap) and ℓ_{p}-(map)

Let us introduce

Definition 6.2.1 Let E ⊆ ℤ and X be a homogeneous Banach space on 𝕋. E enjoys the Fourier block unconditionality property (𝒰) in X whenever, for any ε > 0 and finite F ⊆ E, there is a finite G ⊆ E such that for f ∈ B_{X_{F}} and g ∈ B_{X_{E\G}}

osc_{ϵ∈𝕊} ∥ϵf + g∥_{X} ≤ ε.  (21)

Lemma 6.2.2 Let E ⊆ ℤ and X be a homogeneous Banach space on 𝕋. The following are equivalent.

(i) X_{E} has (u(τ_{f})), where τ_{f} is the topology of pointwise convergence of the Fourier coefficients:

x_{n} →^{τ_{f}} 0 ⟺ ∀k x̂_{n}(k) → 0.

(ii) E enjoys (𝒰) in X. (iii) X_{E} enjoys the property of block unconditionality for any, or equivalently for some, a.s. of multipliers {T_{k}}.

Proof. (i) ⇒ (ii). Suppose that (ii) fails: there are ε > 0 and a finite F such that for each finite G, there are x_{G} ∈ B_{X_{F}} and y_{G} ∈ B_{X_{E\G}} such that

osc_{ϵ∈𝕊} ∥ϵx_{G} + y_{G}∥ > ε.

As B_{X_{F}} is compact, we may suppose x_{G} = x. As y_{G} →^{τ_{f}} 0, (u(τ_{f})) fails. (ii) ⇒ (iii). Let C be a uniform bound for ∥T_{k}∥. Let n ≥ 1 and ε > 0. Let F be the finite spectrum of T_{n}. Let G be such that (21) holds for all f ∈ B_{X_{F}} and g ∈ B_{X_{E\G}}. Now there is a term V in de la Vallée-Poussin’s a.s. such that V |_{X_{G}} = Id|_{X_{G}} and ∥V ∥_{ℒ(X_{E})} ≤ 3. As V has finite rank, we may choose m > n such that ∥(Id − T_{m})V ∥_{ℒ(X_{E})} = ∥V (Id − T_{m})∥_{ℒ(X_{E})} ≤ ε. Let then x ∈ T_{n}B_{X_{E}} and y ∈ (Id − T_{m})B_{X_{E}}. We have

∥ϵx + y∥ ≤ ∥ϵx + (Id − V)y∥ + ε ≤^{(21)} ∥x + (Id − V)y∥ + 4(C + 1)ε + ε
≤ ∥x + y∥ + (4C + 6)ε.

(iii) ⇒ (i) is proved as Lemma 4.2.2 (ii) ⇒ (i): note that if y_{j} →^{τ_{f}} 0, then ∥T y_{j}∥ → 0 for any finite rank multiplier T.

We may now prove the main result of this section.

Theorem 6.2.3 Let E ⊆ ℤ and X be a homogeneous Banach space on 𝕋. If X_{E} has (umap), then E enjoys (𝒰) in X. Conversely, if E enjoys (𝒰) in X and furthermore X_{E} has (uap) and finite cotype, or simply ℓ_{1}-(ap), then X_{E} has (umap). In particular,

(i) For 1 < p < ∞, L_{E}^{p}(𝕋) has (umap) if and only if E enjoys (𝒰) in L^{p}(𝕋). (ii) L_{E}^{1}(𝕋) has (umap) if and only if E enjoys (𝒰) in L^{1}(𝕋) and L_{E}^{1}(𝕋) has (uap). (iii) If E enjoys (𝒰) in 𝒞(𝕋) and 𝒞_{E}(𝕋) has ℓ_{1}-(ap), in particular if E is a Sidon set, then 𝒞_{E}(𝕋) has (umap).

Proof. Notice first that (umap) implies (𝒰) by Lemma 6.2.2 (iii) ⇒ (ii).

(i) Notice that L_{E}^{p}(𝕋) (1 < p < ∞) has an unconditional (fdd) of multipliers {π_{E∩I_{k}}} and cotype 2 ∨ p. Thus (𝒰) implies (umap) by Theorem 4.3.3(ii) ⇒ (iv). By Lemma 6.2.2, part (ii) and (iii) follow from Theorem 4.3.1(ii) ⇒ (iv) and Theorem 4.3.3(ii) ⇒ (iv) respectively.

Remark 6.2.4 Consider the special case E = {0} ∪ {j^{k}}_{k≥0}, |j| ≥ 2, and suppose X_{E} has complex (umap). By Theorem 6.2.3,

osc_{ϵ∈𝕋} ∥ϵa + b e_{j^{k}} +c e_{j^{k+1}} ∥ ⟶_{k→∞} 0.

Let us show that then {0, 1, j} is a 1-unconditional basic sequence in X. Indeed, for any ϵ, μ, ν ∈ 𝕋, and choosing κ such that μκ = νκ^{j},

∥ϵa + μb e_{1} +νc e_{j} ∥ = ∥ϵa + μκb e_{1} +νκ^{j}c e_{j} ∥
= ∥ϵ\overline{μκ}a + b e_{1} +c e_{j} ∥ = ∥ϵ\overline{μκ}a + b e_{j^{k}} +c e_{j^{k+1}} ∥

whose oscillation tends to 0 with k. By Proposition 2.2.1(i), X_{E} fails complex (umap) if X is L^{p}(𝕋), p not an even integer, or 𝒞(𝕋). By Proposition 2.2.1(ii), L_{E}^{2n}(𝕋), n ≥ 1 integer, fails complex (umap) if j is positive and n ≥ j, or if j is negative and n ≥ |j| + 1.

The study of ℓ_{p}-(map) in X_{E} reduces to the trivial case p = 2, X = L^{2}(𝕋), and to the case p = 1, X = 𝒞(𝕋). To see this, note that we have by a repetition of the arguments of Lemma 6.2.2

Lemma 6.2.5 Let E ⊆ ℤ and X be a homogeneous Banach space. The following properties are equivalent.

(i) X_{E} has m_{p}(τ_{f}). (ii) E enjoys the following property ℳ_{p} in X: for any ε > 0 and finite F ⊆ E, there is a finite G ⊆ F such that for f ∈ B_{X_{F}} and g ∈ B_{X_{E\G}}

|∥f + g∥_{X} − (∥f∥_{X}^{p} + ∥g∥_{X}^{p})^{1/p}| ≤ ε

(iii) X_{E} enjoys m_{p}(T_{k}) for any, or equivalently for some, a.s. of multipliers.

Proposition 6.2.6 Let E ⊆ ℤ and X be a homogeneous Banach space.

(i) If X_{E} has ℓ_{p}-(map), then E enjoys ℳ_{p} in X. (ii) L_{E}^{q}(𝕋) has ℓ_{p}-(map) if and only if p = q = 2. (iii) 𝒞_{E}(𝕋) has ℓ_{1}-(map) if and only if it has ℓ_{1}-(ap) and E enjoys ℳ_{1} in 𝒞(𝕋): for all ε > 0 and finite F ⊆ E, there is a finite G ⊆ E such that

∀f ∈ 𝒞_{F} (𝕋) ∀g ∈ 𝒞_{E\G}(𝕋) ∥f∥_{∞} + ∥g∥_{∞} ≤ (1 + ε)∥f + g∥_{∞}.

Proof. (i) Let ε > 0. Let {T_{k}} be an a.s. of multipliers that satisfies (16) with C < 1 + ε. By the argument of [58, Lemma 5], we may assume that the T_{k}’s have their range in 𝒫_{E}(𝕋). Let n ≥ 1 be such that (∑_{k>n} ∥ΔT_{k}f∥_{X}^{p})^{1/p} < ε for f ∈ B_{X_{F}}. Let G be such that T_{k}g = 0 for k ≤ n and g ∈ X_{E\G}. Then successively

|(∑_{k≤n} ∥ΔT_{k}(f + g)∥_{X}^{p})^{1/p} − (∑ ∥ΔT_{k}f∥_{X}^{p})^{1/p}| ≤ ε,

|(∑_{k>n} ∥ΔT_{k}(f + g)∥_{X}^{p})^{1/p} − (∑ ∥ΔT_{k}g∥_{X}^{p})^{1/p}| ≤ ε,

|(∑ ∥ΔT_{k}(f + g)∥_{X}^{p})^{1/p} − (∑ ∥ΔT_{k}f∥_{X}^{p} + ∑ ∥ΔT_{k}g∥_{X}^{p})^{1/p}| ≤ 2^{1/p}ε

and

|∥f + g∥_{X} − (∥f∥_{X}^{p} + ∥g∥_{X}^{p})^{1/p}| ≤ 2ε(1 + 2^{1/p}).

(ii) By Corollary 5.2.2, we necessarily have p = 2. Furthermore, if L_{E}^{q}(𝕋) has ℓ_{2}-(map), then by property ℳ_{2}

|∥ e_{n} + e_{m} ∥_{q} − \sqrt{2}| ⟶_{m→∞} 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 ∥ · ∥_{q} and ∥ · ∥_{2} occurs for almost everywhere constant functions. (iii) Use Theorem 5.3.3.

7 Property (umap) and arithmetical block independence

We may now apply the technique used in the investigation of (umbs) in order to obtain arithmetical conditions analogous to (ℐ_{n}) (see Def. 2.4.1) for (umap). According to Theorem 6.2.3, it suffices to investigate property (𝒰) of block unconditionality: we have to compute an expression of type ∥f + ϵg∥_{p}, where the spectra of f and g are far apart and ϵ ∈ 𝕊. As before, 𝕊 = 𝕋 (vs. 𝕊 = 𝔻) is the complex (vs. real) choice of signs.

7.1 Property of block independence

To this end, we return to the notation of Computational lemmas 2.3.1 and 2.3.4. Define

Ψ_{r}(ϵ, z) = Θ_{r}((\overbrace{1, …, 1}^{j}, \overbrace{ϵ, …, ϵ}^{m−j}), z)

= ∫ |e_{r_{0}}(t) + ∑_{i=1}^{j} z_{i} e_{r_{i}}(t) + ϵ ∑_{i=j+1}^{m} z_{i} e_{r_{i}}(t)|^{p}dm(t)
= ∑_{α∈ℕ^{m}} \binom{p/2}{α}^{2} ∏ |z_{i}|^{2α_{i}} + ∑_{{α≠β∈ℕ^{m} \atop α∼β}} \binom{p/2}{α} \binom{p/2}{β} ϵ^{Σ_{i>j}α_{i}−β_{i}} ∏ z_{i}^{α_{i}} \overline{z_{i}}^{β_{i}}.(22)

As in Computational lemma 2.3.4, we make the following observation:

Computational lemma 7.1.1 Let ζ_{0}, …, ζ_{m} ∈ ℤ^{∗} and γ, δ be as in (4). If the arithmetic relation (5) holds, then the coefficient of the term ϵ^{Σ_{i>j}γ_{i}−δ_{i}} ∏ z_{i}^{γ_{i}} \overline{z_{i}}^{δ_{i}} in (22) is \binom{p/2}{γ} \binom{p/2}{δ} and thus independent of r. If ∑ |ζ_{i}| ≤ p or p is not an even integer, this coefficient is nonzero. If ζ_{0} + … + ζ_{j} is nonzero (vs. odd), then this term is nonconstant in ϵ ∈ 𝕊.

Thus the following arithmetical property shows up. It is similar to property (ℐ_{n}) of almost independence.

Definition 7.1.2 Let E ⊆ ℤ and n ≥ 1.

(i) E enjoys the complex (vs. real) property (𝒥_{n}) of block independence if for any ζ ∈ Z_{n}^{m} with ζ_{1} + … + ζ_{j} nonzero (vs. odd) and given p_{1}, …, p_{j} ∈ E, there is a finite G ⊆ E such that ζ_{1}p_{1} + … + ζ_{m}p_{m} ≠ 0 for all p_{j+1}, …, p_{m} ∈ E \ G. (ii) E enjoys exactly complex (vs. real) (𝒥_{n}) if furthermore it fails complex (vs. real) (𝒥_{n+1}). (iii) E enjoys complex (vs. real) (𝒥_{∞}) if it enjoys complex (vs. real) (𝒥_{n}) for all n ≥ 1.

The complex (vs. real) property (𝒥_{n}) means precisely the following. “For every finite F ⊆ E there is a finite G ⊆ E such that for any two representations of any k ∈ ℤ as a sum of n elements of F ∪ (E \ G)

p_{1} + … + p_{n} = k = p′_{1} + … + p′_{n}

one necessarily has

{j : p_{j} ∈ F } = {j : p′_{j} ∈ F } in ℤ (vs. in ℤ/2ℤ).”

Thus property (𝒥_{n}) has, unlike (ℐ_{n}), a complex and a real version. Real (𝒥_{n}) is strictly weaker than complex (𝒥_{n}): see Section 8. Notice that (𝒥_{1}) is void and (𝒥_{n+1}) ⇒ (𝒥_{n}) in both complex and real cases. Also (ℐ_{n}) ⇏ (𝒥_{n}): we shall see in the following section that E = {0} ∪ {n^{k}}_{k≥0} provides a counterexample. The property (𝒥_{2}) of real block independence appears implicitly in [58, Lemma 12].

Remark 7.1.3 In spite of the intricate form of this arithmetical property, (𝒥_{n}) is the “simplest” candidate, in some sense, that reflects the features of (𝒰): it must hold for a set E if and only if it holds for a translate E + k of this set: this explains ∑ ζ_{i} = 0 in Definition 7.1.2(i); as for the property (𝒰) of block independence, it must connect the break of E with its tail; Li gives an example of a set E whose pace does not tend to infinity while 𝒞_{E}(𝕋) has ℓ_{1}-(map). 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 ζ_{1} + … + ζ_{j} be nonzero (vs. odd) in Definition 7.1.2(i).

We now repeat the argument of Theorem 2.4.2 to obtain an analogous statement which relates property (𝒰) of Definition 6.2.1 with our new arithmetical conditions

Lemma 7.1.4 Let E = {n_{k}} ⊆ ℤ and 1 ≤ p < ∞.

(i) Suppose p is an even integer. Then E enjoys the complex (vs. real) Fourier block unconditionality property (𝒰) in L^{p}(𝕋) if and only if E enjoys complex (vs. real) (𝒥_{p/2}). (ii) If p is not an even integer and E enjoys complex (vs. real) (𝒰) in L^{p}(𝕋), then E enjoys complex (vs. real) (𝒥_{∞}).

Proof. Let us first prove the necessity of the arithmetical property and assume E fails (𝒥_{n}): then there are ζ_{0}, …, ζ_{m} ∈ ℤ^{∗} with ∑ ζ_{i} = 0, ∑ |ζ_{i}| ≤ 2n and ζ_{0} +…+ζ_{j} nonzero (vs. odd); there are r_{0}, …, r_{j} ∈ E and sequences r_{j+1}^{l}, …, r_{m}^{l} ∈ E \ {n_{1}, …, n_{l}} such that

ζ_{0}r_{0} + … + ζ_{j}r_{j} + ζ_{j+1}r_{j+1}^{l} + … + ζ_{m}r_{m}^{l} = 0.

Assume E enjoys (𝒰) in L^{p}(𝕋). Then the oscillation of Ψ_{r} in (22) satisfies

osc_{ϵ∈𝕊} Ψ_{r^{l}}(ϵ, z) ⟶_{l→∞} 0  (23)

for each z ∈ D^{m}. The argument is now exactly the same as in Theorem 2.4.2: we may assume that the sequence of functions Ψ_{r^{l}} converges in 𝒞^{∞}(𝕊 × D^{m}) to a function Ψ. Then by (23), Ψ(ϵ, z) is constant in ϵ for each z ∈ D^{m}, and this is impossible by Computational lemma 7.1.1 if p is either not an even integer or p ≥ 2n. Let us now prove the sufficiency of (𝒥_{p/2}) when p is an even integer. First, let A_{n}^{k,l} = {α ∈ A_{n} : α_{i} = 0 for k < i ≤ l} (A_{n} is defined before Prop. 2.2.1), and convince yourself that (𝒥_{p/2}) is equivalent to

∀k ∃l ≥ k ∀α, β ∈ A_{p/2}^{k,l} ∑ α_{i}n_{i} = ∑ β_{i}n_{i} ⇒ ∑_{i≤k} α_{i} = ∑_{i≤k} β_{i} (vs. mod 2). (24)

Let f = ∑ a_{i} e_{n_{i}} ∈ 𝒫_{E}(𝕋). Let k ≥ 1 and ϵ ∈ 𝕊. By the multinomial formula,

∥ϵπ_{k}f + (f − π_{l}f)∥_{p}^{p} = ∫ | ∑_{α∈A_{p/2}^{k,l}} \binom{p/2}{α} ϵ^{Σ_{p≤k}α_{i}}(∏ a_{i}^{α_{i}}) e_{Σα_{i}n_{i}} |^{2}dm

= ∫ |∑_{j=0}^{n} ϵ^{j} ∑_{{α∈A_{p/2}^{k,l} \atop α_{1}+...+α_{k}=j}} \binom{p/2}{α} (∏ a_{i}^{α_{i}}) e_{Σα_{i}n_{i}} |^{2}dm.

(24) now signifies that we may choose l ≥ 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 ∥ϵπ_{k}f + (f − π_{l}f)∥_{p} is constant for ϵ ∈ 𝕊 and E enjoys (𝒰) in L^{p}(𝕋).

Note that for even p, we have as in Proposition 2.2.1 a constant C_{p} > 1 such that either (21) holds for ε = 0 or fails for any ε ≤ C_{p}. We thus get

Corollary 7.1.5 Let E ⊆ ℤ and p be an even integer. If E enjoys complex (vs. real) (𝒰) in L^{p}(𝕋), then there is a partition E = ⋃ E_{k} into finite sets such that for any coarser partition E = ⋃ E′_{k}

∀f ∈ 𝒫_{E}(𝕋) osc_{ϵ_{k}∈𝕊}∥∑ ϵ_{k}π_{E′_{2k}}f∥_{p} = 0

Among other consequences, E = E_{1} ∪ E_{2} where the L_{E_{i}}^{p}(𝕋) have a complex (vs. real) 1-unconditional (fdd).

Question 7.1.6 Is this rigidity proper to translation invariant subspaces of L^{p}(𝕋) with p an even integer, or generic for all its subspaces (see [13]) ?

7.2 Main result

Lemma 7.1.4 and Theorem 6.2.3 yield the main result of this section.

Theorem 7.2.1 Let E ⊆ ℤ and 1 ≤ p < ∞.

(i) Suppose p is an even integer. Then L_{E}^{p}(𝕋) has complex (vs. real) (umap) if and only if E enjoys complex (vs. real) (𝒥_{p/2}). (ii) If p is not an even integer and L_{E}^{p}(𝕋) has complex (vs. real) (umap), then E enjoys complex (vs. real) (𝒥_{∞}).

Corollary 7.2.2 Let E ⊆ ℤ.

(i) If 𝒞_{E}(𝕋) has complex (vs. real) (umap), then E enjoys complex (vs. real) (𝒥_{∞}). (ii) If any L_{E}^{p}(𝕋), p not an even integer, has complex (vs. real) (umap), then all L_{E}^{p}(𝕋) with p an even integer have complex (vs. real) (umap).

Suppose p is an even integer. Then Section 8 gives various examples of sets such that L_{E}^{p}(𝕋) has complex or real (umap). Proposition 9.2.1 gives a general growth condition that ensures (umap). For X = L^{p}(𝕋), p not an even integer, and X = 𝒞(𝕋), however, we encounter the same obstacle as for (umbs). Section 8 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 𝒞_{E}(𝕋) — and thus by [58, Th. 7] all L_{E}^{p}(𝕋) (1 ≤ p < ∞) as well — have (umap). Sets found by Li [58]: Kronecker’s theorem is used to construct a set containing arbitrarily long arithmetic sequences and a set whose pace does not tend to infinity. Meyer’s [68, VIII] techniques are used to construct a Hilbert set. The sets that satisfy the growth condition of Theorem 9.3.1; Sequences E = {n_{k}} ⊆ ℤ such that n_{k+1}/n_{k} is an odd integer: see Proposition 9.1.1.

Question 7.2.3 We know no example of a set E such that some L_{E}^{p}(𝕋), p not an even integer, has (umap) while 𝒞_{E}(𝕋) fails it.

There is also a good arithmetical description of the case where {π_{k}} or a subsequence thereof realizes (umap).

Proposition 7.2.4 Let E = {n_{k}}_{k≥1} ⊆ ℤ. Consider a partition E = ⋃_{k≥1} E_{k} into finite sets.

(i) Suppose p is an even integer. The series ∑ π_{E_{k}} realizes complex (vs. real) (umap) in L_{E}^{p}(𝕋) if and only if there is an l ≥ 1 such that

\{ \begin{matrix}p_{1}, …, p_{m} ∈ E \\ ζ_{1}p_{1} + … + ζ_{m}p_{m} = 0 \end{matrix} ⇒ ∀k ≥ l ∑_{p_{j}∈E_{k}} ζ_{j} = 0 (vs. is even)  (25)

for all ζ ∈ Z_{p/2}^{m}. Then L_{E}^{p}(𝕋) admits the series π_{∪_{k<l}E_{k}} + ∑_{k≥l} π_{E_{k}} as 1-uncondi-tional (fdd). In particular, choose E_{k} = {n_{k}}. The sequence {π_{k}} realizes complex and real (umap) in L_{E}^{p}(𝕋) if and only if there is a finite G such that for ζ ∈ Z_{p/2}^{m}

\{ \begin{matrix}p_{1}, …, p_{m} ∈ E \\ ζ_{1}p_{1} + … + ζ_{m}p_{m} = 0 \end{matrix} ⇒ p_{1}, …, p_{m} ∈ G.  (26)

Then E \ G is a 1-(ubs) and E enjoys (ℐ_{p/2}). (ii) Suppose p is not an even integer. If ∑ π_{E_{k}} realizes complex (vs. real) (umap) in L_{E}^{p}(𝕋), then for each ζ ∈ Z^{m} there is an l ≥ 1 such that (25) holds. In particular, if {π_{k}} realizes either complex or real (umap) in L_{E}^{p}(𝕋), then for all ζ ∈ Z^{m} there is a finite G such that (26) holds. This is equivalent to (ℐ_{∞}).

Proof. It is analogous to the proof of Lemma 7.1.4: suppose we have ζ ∈ Z_{n}^{m} such that (25) fails for any l ≥ 1. Then there are ζ_{0}, …, ζ_{m} ∈ ℤ^{∗} with ∑ ζ_{i} = 0, ∑ |ζ_{i}| ≤ 2n and ζ_{0} + … + ζ_{j} nonzero (vs. odd) for some j; for each l, there are r_{0}^{l}, …, r_{j}^{l} ∈ ∪_{k<l}E_{k} and r_{j+1}^{l}, …, r_{m}^{l} ∈ ∪_{k≥l}E_{k} such that ζ_{0}r_{0}^{l} + … + ζ_{m}r_{m}^{l} = 0. But then ∑ π_{E_{k}} cannot realize complex (vs. real) (umap): the function Ψ_{r} in (22) would satisfy (23) and we would obtain a contradiction as in Theorem 2.4.2. Sufficiency in (i) and (i′) is proved exactly as in Lemma 7.1.4(i).

In particular, suppose that the cardinal E_{k} is uniformly bounded by M and {π_{E_{k}}} realizes (umap) in L_{E}^{p}(𝕋). If p ≠ 2 is an even integer, then E is a Λ(p) set as union of a finite set and M p/2-independent sets (see Prop. 2.2.1 and [86, Th. 4.5(b)]). If p is not an even integer, then E is a Λ(q) set for all q by the same argument.

8 Examples for (umap): block independent sets of characters

8.1 General properties

The pairing ⟨ζ, E⟩ underlines the asymptotic nature of property (𝒥_{n}). It has been defined before Proposition 3.1.1, whose proof adapts to

Proposition 8.1.1 Let E = {n_{k}} ⊆ ℤ.

(i) If ⟨ζ, E⟩ < ∞ for ζ_{1}, …, ζ_{m} ∈ ℤ^{∗} with ∑ ζ_{i} nonzero (vs. odd), then E fails complex (vs. real) (𝒥_{|ζ_{1}|+...+|ζ_{m}|}). Conversely, if E fails complex (vs. real) (𝒥_{n}), then there are ζ_{1}, …, ζ_{m} ∈ ℤ^{∗} with ∑ ζ_{i} nonzero (vs. odd) and ∑ |ζ_{i}| ≤ 2n−1 such that ⟨ζ, E⟩ < ∞. (ii) Thus E enjoys complex (vs. real) (𝒥_{∞}) if and only if ⟨ζ, E⟩ = ∞ for all ζ_{1}, …, ζ_{m} ∈ ℤ^{∗} with ∑ ζ_{i} nonzero (vs. odd).

Proof of the converse in (i). If E fails complex (vs. real) (𝒥_{n}), then there are ζ ∈ Z_{n}^{m} with ζ_{1} + … + ζ_{j} nonzero (vs. odd), p_{1}, …, p_{j} ∈ E and sequences p_{j+1}^{l}, …, p_{m}^{l} ∈ {n_{k}}_{k≥l} such that ∑_{i>j} ζ_{i}p_{i}^{l} = − ∑_{i≤j} ζ_{i}p_{i}. Let ζ′ = (ζ_{j+1}, …, ζ_{m}). Then ∑ |ζ′_{i}| ≤ 2n − 1 and ⟨ζ′, E⟩ < ∞.

An immediate application is, as in Proposition 3.1.1,

Proposition 8.1.2 Let E = {n_{k}} ⊆ ℤ.

(i) Suppose E enjoys (ℐ_{2n−1}). Then E enjoys complex (𝒥_{n}) and actually there is a finite set G such that (26) holds for ζ ∈ Z_{n}^{m}.

(ii) Suppose E enjoys (ℐ_{∞}). Then E enjoys complex (𝒥_{∞}) and actually for all ζ ∈ Z^{m} there is a finite G such that (26) holds. (iii) Complex and real (𝒥_{∞}) are stable under bounded perturbations of E. (iv) Suppose there is h ∈ ℤ such that E ∪ {h} fails complex (vs. real) (𝒥_{n}). Then E fails complex (vs. real) (𝒥_{2n−1}). Thus the complex and real properties (𝒥_{∞}) are stable under unions with an element: if E enjoys it, then so does E ∪ {h}.

(v) Suppose jF + s, kF + t ∈ E for an infinite F, j ≠ k ∈ ℤ^{∗} and s, t ∈ ℤ. Then E fails complex (𝒥_{|j|+|k|}), and also real (𝒥_{|j|+|k|}) if j and k have different parity.

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

8.2 Geometric sequences

Let G = {j^{k}}_{k≥0} with j ∈ ℤ \ {−1, 0, 1}. We resume Remark 6.2.4.

(1) As G, jG ⊆ G, G fails complex (𝒥_{|j|+1}), and also real (𝒥_{|j|+1}) if j is even. The solutions (8) to the Diophantine equation (7) show at once that G enjoys complex (𝒥_{|j|}), since there is no arithmetical relation ζ ∈ Z_{|j|}^{m} between the break and the tail of G. If j is odd, then G enjoys in fact real (𝒥_{∞}). Indeed, let ζ_{1}, …, ζ_{m} ∈ ℤ^{∗} and k_{1} < … < k_{m}: then ∑ ζ_{i}j^{k_{i}} ∈ j^{k_{1}}ℤ and either | ∑ ζ_{i}j^{k_{i}} | ≥ j^{k_{1}} or ∑ ζ_{i}j^{k_{i}} = 0. Thus, if ⟨ζ, E⟩ < ∞ then ⟨ζ, E⟩ = 0 and ∑ ζ_{i} is even since j is odd. Now apply Proposition 8.1.1(iii). The same argument yields that even G ∪ −G ∪ {0} enjoys real (𝒥_{∞}). Actually much more is true: see Proposition 9.1.1.

(2) G∪{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 (8) may be written as (−j+1)·0+j·j^{k} +(−1)·j^{k+1} = 0. If j is positive, (−j+1)+j+(−1) ≤ 2j and G ∪ {0} fails complex (𝒥_{j}). A look at (8) shows that it nevertheless enjoys complex (𝒥_{j−1}). On the other hand, G ∪ {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 ∪ {0} still enjoys real (𝒥_{∞}) if j is odd.

8.3 Symmetric sets

By Proposition 3.1.1(iii) and 8.1.2(vi), they do enjoy neither (ℐ_{2}) nor complex (𝒥_{2}). They may nevertheless enjoy real (𝒥_{n}). Introduce property (𝒥n^{sym}) for E: it holds if for all p_{1}, …, p_{j} ∈ E and η ∈ ℤ^{∗}^{m} with ∑_{1}^{m} η_{i} even, ∑_{1}^{m} |η_{i}| ≤ 2n and η_{1} + … + η_{j} odd, there is a finite set G such that η_{1}p_{1} + … + η_{m}p_{m} ≠ 0 for any p_{j+1}, …, p_{m} ∈ E \ G. Then we obtain

Proposition 8.3.1 E ∪ −E has real (𝒥_{n}) if and only if E has (𝒥n^{sym}).

Proof. By definition, E ∪ −E has real (𝒥_{n}) if and only if for all p_{1}, …, p_{j} ∈ E and ζ^{1}, ζ^{2} ∈ ℤ^{m} with ζ^{1} + ζ^{2} ∈ Z_{n}^{m} and odd ∑_{i≤k} ζ_{i}^{1} − ζ_{i}^{2}, there is a finite set G such that ∑(ζ_{i}^{1} − ζ_{i}^{2})p_{i} ≠ 0 for any p_{j+1}, …, p_{m} ∈ E \ G — and thus if and only if E enjoys (𝒥n^{sym}): just consider the mappings between arithmetical relations (ζ^{1}, ζ^{2}) ↦ η = ζ^{1} − ζ^{2} and η ↦ (ζ^{1}, ζ^{2}) such that η = ζ^{1} − ζ^{2}, where ζ_{i}^{1} = η_{i}/2 if η_{i} is even and, noting that the number of odd η_{i}’s must be even, ζ_{i}^{1} = (η_{i} − 1)/2 and ζ_{i}^{1} = (η_{i} + 1)/2 respectively for each half of them. Consider again a geometric sequence G = {j^{k}} with j ≥ 2. If j is odd, we saw before that G ∪ −G and G ∪ −G ∪ {0} enjoy real (𝒥_{∞}). If j is even, then G ∪ −G fails real (𝒥_{j+1}) since G does. G ∪ −G ∪ {0} fails real (𝒥_{j/2+1}) by the arithmetical relation 1 · 0 + j · j^{k} + (−1) · j^{k+1} = 0 and Proposition 8.3.1. G ∪ −G enjoys real (𝒥_{j}) and G ∪ −G ∪ {0} enjoys real (𝒥_{j/2}) as the solutions in (8) show by a simple checking.

8.4 Algebraic and transcendental numbers

The proof of Proposition 3.3.1 adapts to

Proposition 8.4.1 Let E = {n_{k}} ⊆ ℤ.

(i) If n_{k+1}/n_{k} → σ where σ > 1 is transcendental, then E enjoys complex (𝒥_{∞}). (ii) Let n_{k} = [σ^{k}] with σ > 1 algebraic. Let P (x) = ζ_{0} + … + ζ_{d}x^{d} be the corresponding polynomial of minimal degree. Then E fails complex (𝒥_{|ζ_{0}|+...+|ζ_{d}|}), and also real (𝒥_{|ζ_{0}|+...+|ζ_{d}|}) if P (1) is odd.

8.5 Polynomial sequences

Let E = {P (k)} for a polynomial P of degree d. The arithmetical relation (9) does not adapt to property (𝒥_{n}). Notice, though, that {Δ^{j}P }_{j=1}^{d} is a basis for the space of polynomials of degree less than d and that 2^{d}P (k) − P (2k) is a polynomial of degree at most d − 1. Writing it in the basis {Δ^{j}P }_{1}^{d} yields an arithmetical relation 2^{d} · P (k) − 1 · P (2k) + ∑_{j=0}^{d} ζ_{j} · P (k − j) = 0 such that 2^{d} − 1 + ∑ ζ_{j} is odd. By Proposition 8.1.1 (ii), E fails real (𝒥_{n}) for a certain n. This n may be bounded in certain cases: The set of squares fails real (𝒥_{2}): let F_{n} be the Fibonacci 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} → ∞ and F_{n}F_{n+2} − F_{n+1}^{2} = (−1)^{n} (see [26]). Inspired by [71, p. 15], we observe that

(F_{n}F_{n+2} + F_{n+1}^{2})^{2} + (F_{n+1}^{2})^{2} = (F_{n}F_{n+1} + F_{n+1}F_{n+2})^{2} + 1^{2}

The set of cubes fails real (𝒥_{2}): starting from Binet’s [5] simplified solution of Euler’s equation [24], 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. The set of biquadrates fails real (𝒥_{3}): by an equality of Ramanujan (see [81, p. 386]),

(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 would imply that the set of kth powers has complex (𝒥_{2}) for k ≥ 5.

Conclusion By Theorem 7.2.1, property (𝒥_{n}) yields directly (umap) in the space L^{2p}(𝕋), p ≤ n integer. But we do not know whether (𝒥_{∞}) ensures (umap) in spaces L^{p}(𝕋), p not an even integer, or 𝒞(𝕋). Nevertheless, the study of property (𝒥_{3}) permits us to determine the density of sets such that X_{E} enjoys (umap) for some X ≠ L^{2}(𝕋), L^{4}(𝕋): see Proposition 11.2. Other applications are given in Section 13.

9 Positive results: parity and a sufficient growth condition

9.1 𝒞_{{3^{k}}}(𝕋) has real (umap) because 3 is odd

In the real case, parity plays an unexpected rôle.

Proposition 9.1.1 Let E = {n_{k}} ⊆ ℤ and suppose that n_{k+1}/n_{k} is an odd integer for all sufficiently large k. Then 𝒞_{E}(𝕋) has real (umap).

Then X_{E} also has real (umap) for every homogeneous Banach space X on 𝕋.

Proof. Let us verify that real (𝒰) holds. Let ε > 0 and F ⊆ E ∩[−n, n]. Let l, to be chosen later, such that n_{k+1}/n_{k} is an odd integer for k ≥ l. Take G ⊇ {n_{1}, …, n_{l}} finite. Let f ∈ B_{𝒞_{F}} and g ∈ B_{𝒞_{E\G}} . Then g(u exp iπ/n_{l}) = −g(u) and

|f(u exp iπ/n_{l}) − f(u)| ≤ π/|n_{l}| · ∥f′∥_{∞} ≤ πn/|n_{l}| ≤ ε

by Bernstein’s inequality and for l large enough. Thus, for some u ∈ 𝕋,

∥f − g∥_{∞} = |f(u) + g(u exp iπ/n_{l})|
≤ |f(u exp iπ/n_{l}) + g(u exp iπ/n_{l})| + ε
≤ ∥f + g∥_{∞} + ε.

As E is a Sidon set, we may apply Theorem 6.2.3(iii).

Furthermore, if E satisfies the hypothesis of Proposition 9.1.1, so does E ∪ −E = {n_{1}, −n_{1}, n_{2}, −n_{2}, …}. But E ∪ −E fails even complex (𝒥_{2}) and no X_{E∪−E} ≠ L_{E∪−E}^{2}(𝕋) 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 8.1.2(vi).

Remark 9.1.2 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 𝒞_{E_{i}}(𝕋): 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}) · n_{k} = 0 and we may apply Proposition 7.2.4(ii). This justifies the use of Theorem 6.2.3(iii).

9.2 Growth conditions: the case L^{p}(𝕋), p an even integer

For X = L^{p}(𝕋) with p an even integer, a look at (ℐ_{n}) and (𝒥_{n}) gives by Theorems 2.4.2 and 7.2.1 the following general growth condition:

Proposition 9.2.1 Let E = {n_{k}} ⊆ ℤ and p ≥ 1 an integer. If

lim inf |n_{k+1}/n_{k}| ≥ p + 1,  (27)

then {π_{k}} realizes the complex (umap) in L_{E}^{2p}(𝕋) and there is a finite G ⊆ E such that E \ G is a 1-unconditional basic sequence in L^{2p}(𝕋).

Proof. Suppose we have an arithmetical relation

ζ_{1}n_{k_{1}} + … + ζ_{m}n_{k_{m}} = 0 with ζ ∈ Z_{p}^{m} and |n_{k_{1}}| < … < |n_{k_{m}}|.  (28)

Then |ζ_{m}n_{k_{m}}| ≤ |ζ_{1}n_{k_{1}}| + … + |ζ_{m−1}n_{k_{m−1}}|. The left hand side is smallest when |ζ_{m}| = 1. As |ζ_{1}| + … + |ζ_{m}| ≤ 2p and necessarily |ζ_{i}| ≤ p, the right hand side is largest when |ζ_{m−1}| = p and |ζ_{m−2}| = p − 1. Furthermore, it is largest when k_{m} = k_{m−1} + 1 = k_{m−2} + 2. Thus, if (28) holds, then

|n_{k_{m}}| ≤ p|n_{k_{m−1}}| + (p − 1)|n_{k_{m−2}}|.

By (27), this is impossible as soon as m is chosen sufficiently large, because p + 1 > p + (p − 1)/(p + 1).

Note that Proposition 9.2.1 is best possible: if j is negative, then {j^{k}} fails (ℐ_{|j|}). If j is positive, then {j^{k}} ∪ {0} fails complex (𝒥_{j}).

9.3 A general growth condition

Although we could prove that E enjoys (ℐ_{∞}) and (𝒥_{∞}) when n_{k+1}/n_{k} → ∞, we need a direct argument in order to get the corresponding functional properties: we have

Theorem 9.3.1 Let E = {n_{k}} ⊆ ℤ such that n_{k+1}/n_{k} → ∞. Then 𝒞_{E}(𝕋) has ℓ_{1}- (map) with {π_{k}} and E is a Sidon set with constant asymptotically 1. If the ratios n_{k+1}/n_{k} are all integers, then the converse holds.

Note that by Proposition 2.1.3(ii), E is a metric unconditional basic sequence in every homogeneous Banach space X on 𝕋. Further X_{E} has complex (umap) since 𝒞_{E}(𝕋) does.

Proof. Suppose |n_{j+1}/n_{j}| ≥ q for j ≥ l and some q > 1 to be fixed later. Let f = ∑ a_{j} e_{n_{j}} ∈ 𝒫_{E}(𝕋) and k ≥ l. We show by induction that for all p ≥ k

∥π_{p}f∥_{∞} ≥ (1 − \frac{π^{2}}{2} \frac{1 − q^{2(k−p)}}{q^{2} − 1} )∥π_{k}f∥_{∞} + ∑_{j=k+1}^{p}(1 − \frac{π^{2}}{2} \frac{1 − q^{2(j−p)}}{q^{2} − 1} )|a_{j}|. (29)

There is nothing to show for p = k. By Bernstein’s inequality applied to π_{k}f′′ and separately to each a_{j} e′′_{n_{j}}, j > k,

∥π_{p}f′′∥_{∞} ≤ n_{k}^{2}∥π_{k}f∥_{∞} + ∑_{j=k+1}^{p} n_{j}^{2}|a_{j}|.  (30)

Furthermore, by Lemmas 1 and 2 of [68, §VIII.4.2],

∥π_{p+1}f∥_{∞} ≥ ∥π_{p}f∥_{∞} + |a_{p+1}| − π^{2}/(2n_{p+1}^{2})∥π_{p}f′′∥_{∞}.  (31)

(31) together with (29) and (30) yield (29) with p replaced by p + 1. Therefore

∥f∥_{∞} = lim_{p→∞} ∥π_{p}f∥_{∞} ≥ (1 − \frac{π^{2}}{2} \frac{1}{q^{2} − 1})(∥π_{k}f∥_{∞} + ∑_{j=k+1}^{∞} |a_{j}|). (32)

Thus {π_{j}}_{j≥k} realizes ℓ_{1}-(ap) with constant 1 + π^{2}/(2q^{2} − 2 − π^{2}). As q may be chosen arbitrarily large, E has ℓ_{1}-(map) with {π_{j}}. Additionally (32) shows by choosing π_{k}f = 0 that E is a (umbs) in 𝒞(𝕋). Finally, the converse holds by Proposition 8.1.2(vi): if n_{k+1}/n_{k} does not tend to infinity while being integer, then there are h ∈ ℤ\{0, 1} and an infinite F such that F, hF ⊆ E.

Remark 9.3.2 The technique of Riesz products as exposed in [45, Appendix V, §1.II] would have sufficed to prove Theorem 9.3.1.

Remark 9.3.3 Suppose still that E = {n_{k}} ⊆ ℤ with n_{k+1}/n_{k} → ∞. A variation of the above argument yields that the space of real functions with spectrum in E ∪ −E has ℓ_{1}-(ap).

Remark 9.3.4 Note however that there are sets E that satisfy n_{k+1}/n_{k} → 1 and nevertheless enjoy (ℐ_{∞}) (see end of Section 11): they might be (umbs) in 𝒞(𝕋), but this is unknown.

10 An excursion: estimation of the Sidon constant

The proof of Theorem 9.3.1 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.

10.1 Sidon constant of sets with three elements

We can compute explicitly the Sidon constant of {n_{1}, n_{2}, n_{3}}. It is equal to

max_{r,s>0,ϑ∈𝕋}(1 + r + s)/∥1 + r e_{k} +ϑs e_{l} ∥_{∞} with \{ \begin{matrix}k = n_{2} − n_{1} \\ l = n_{3} − n_{1}. \end{matrix}  (33)

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_{i}t_{i} with r_{i} ≥ 0 and t_{i} ∈ 𝕋. Then ∥f∥_{∞} is equal to

r_{1} ∥1 + r_{1}^{−1}r_{2} e_{n_{2}−n_{1}} +ϑr_{1}^{−1}r_{3} e_{n_{3}−n_{1}} ∥_{∞} where ϑ = t′_{1}t′_{2}t_{3} and

\{ \begin{matrix}t′_{1} & is any (n_{3} − n_{2})/(n_{2} − n_{1})-th power of t_{1} \\ t′_{2} & is any (n_{1} − n_{3})/(n_{2} − n_{1})-th power of t_{2}. \end{matrix} (34)

Let us first establish Lemma 10.1.1 Let (a_{ij}) ∈ ℝ^{2} × ℝ^{3} and α, β ∈ ℝ. Consider

\{ \begin{matrix}a_{11} sin α + a_{12} sin β + a_{13} sin(β − α) = 0 \\ a_{21} sin α + a_{22} sin β + a_{23} sin(β − α) = 0. \end{matrix}  (35)

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_{1}d_{2}d_{3} ≠ 0, then the solutions to (S) are α ≡ β ≡ 0 mod. π and, if furthermore |d_{1}| ≤ |d_{2}|+|d_{3}|, |d_{2}| ≤ |d_{1}| + |d_{3}| and |d_{3}| ≤ |d_{1}| + |d_{2}|,

cos α = \frac{d_{3}^{2} − d_{1}^{2} − d_{2}^{2}}{2d_{1}d_{2}} , cos β = \frac{d_{2}^{2} − d_{1}^{2} − d_{3}^{2}}{2d_{1}d_{3}} , cos(β − α) = \frac{d_{1}^{2} − d_{2}^{2} − d_{3}^{2}}{2d_{2}d_{3}} , (36)

where the signs of α and β satisfy d_{2} sin α + d_{3} sin β = 0.

Proof. Suppose that α ≢ 0 or β ≢ 0 mod. π. As (35) is equivalent to

(a_{i1} − a_{i3} cos β) sin α + (a_{i2} + a_{i3} cos α) sin β = 0 : i = 1, 2,  (37)
(a_{11} − a_{13} cos β)(a_{22} + a_{23} cos α) = (a_{21} − a_{13} cos β)(a_{12} + a_{13} cos α).

This simplifies to

d_{1} + d_{2} cos α + d_{3} cos β = 0  (38)

and by (37),

\{ \begin{matrix}(d_{2} sin α + d_{3} sin β)(a_{i2} + a_{i3} cos α) = 0 \\ (d_{2} sin α + d_{3} sin β)(a_{i1} − a_{i3} cos α) = 0 \end{matrix} : i = 1, 2.

Therefore

d_{2} sin α + d_{3} sin β = 0 :  (39)

otherwise d_{1} = d_{2} = d_{3} = 0. Equations (38) and (39) yield solution (36).

Lemma 10.1.2 Let r, s > 0, k, l ∈ ℤ^{∗} distinct and coprime. Let

Φ(t, ϑ) = |1 + r e^{ikt} + e^{iϑ} s e^{ilt} |^{2}
= 1 + r^{2} + s^{2} + 2r cos(kt) + 2s cos(lt + ϑ) + 2rs cos((l − k)t + ϑ).

Let Φ^{∗}(ϑ) = max_{0≤t<2π} Φ(t, ϑ). Then min_{0≤ϑ<2π} Φ^{∗}(ϑ) = Φ^{∗}(π/k).

Proof. Let us first locate the extremal points of Φ. ∇Φ(t, ϑ) = 0 is Equation (35) with

(a_{ij}) = ( \begin{matrix}rk & sl & rs(l − k) \\ 0 & s & rs \end{matrix} ) and \{ \begin{matrix}α = kt \\ β = lt + ϑ. \end{matrix}

In the notation of Lemma 10.1.1, d_{1} = rsk ≠ 0, d_{2} = r^{2}sk ≠ 0, d_{3} = rs^{2}k ≠ 0. The solution (36), if it exists, yields Φ(t, ϑ) = 0 and corresponds to the absolute minimum of Φ. Every other extremal point of Φ satisfies

kt ≡ lt + ϑ ≡ 0 mod. π.

Φ^{∗} is continuous (see [80, Chapter 5.4]) and (2π/k)-periodical: choose j ∈ ℤ such that jl ≡ 1 mod. k. Then

Φ(t + 2jπ/k, ϑ) = |1 + r e^{ikt} +s e^{i(ϑ+2πjl/k)} e^{ilt} |^{2} = Φ(t, ϑ + 2π/k).

Furthermore Φ(t, −ϑ) = Φ(−t, ϑ) and Φ^{∗} is even. Thus Φ^{∗} attains its minimum on [0, π/k]. Let us show that Φ^{∗} has a local minimum in π/k for all values of r, s > 0 except eventually one value of s for a given r. Let t^{∗} be such that Φ(t^{∗}, π/k) = Φ^{∗}(π/k). As before, for j such that jl ≡ 1 mod. k,

Φ(t^{∗}, π/k + ϑ) = Φ(−t^{∗}, −π/k − ϑ) = Φ(−t^{∗} + 2jπ/k, π/k − ϑ).

If ∂Φ/∂ϑ(t^{∗}, π/k) ≠ 0, this shows that Φ^{∗} has a local minimum in π/k. Let us suppose that ∂Φ/∂ϑ(t^{∗}, π/k) = 0. Then ∇Φ(t^{∗}, π/k) = 0 and therefore kt^{∗} = jπ and lt^{∗} + π/k = j′π for some j, j′ ∈ ℤ. Then j or j′ must be odd. We have

\frac{∂^{2}Φ}{∂t^{2}}(t^{∗}, π/k) = −2(rk^{2}(−1)^{j} + sl^{2}(−1)^{j′} + rs(l − k)^{2}(−1)^{j+j′}).  (40)

We shall suppose that ∂^{2}Φ/∂t^{2}(t^{∗}, π/k) ≠ 0, 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 π/k such that

t_{∗}(π/k) = t^{∗} , ∂Φ/∂t(t_{∗}(ϑ), ϑ) = 0 , ∂^{2}Φ/∂t^{2}(t_{∗}(ϑ), ϑ) < 0.

Let Φ_{∗}(ϑ) = Φ(t_{∗}(ϑ), ϑ). Then we have Φ′_{∗}(π/k) = ∂Φ/∂ϑ(t^{∗}, π/k) = 0 and a computation yields

Φ′′_{∗}(π/k) = \frac{\frac{∂^{2}Φ}{∂t^{2}} \frac{∂^{2}Φ}{∂ϑ^{2}} − \frac{∂^{2}Φ}{∂t∂ϑ}^{2}}{\frac{∂^{2}Φ}{∂t^{2}}} (t^{∗}, π/k) = 4rsk^{2} \frac{∂^{2}Φ}{∂t^{2}}(t^{∗}, π/k)^{−1}Δ,

where Δ = ((−1)^{j+j′} + r(−1)^{j′} + s(−1)^{j}). Let us prove that Δ < 0 and thus Φ′′_{∗}(π/k) > 0 and that therefore Φ_{∗} and consequently Φ^{∗} have a local minimum in π/k. If we had Δ ≥ 0 and j even, j′ odd: then −1 − r + s ≥ 0 and by (40)

\frac{∂^{2}Φ}{∂t^{2}}(t^{∗}, π/k) ≥ 2(−rk^{2} + (1 + r)l^{2} + r(1 + r)(l − k)^{2}) = 2(r(l − k) + l)^{2} ≥ 0;

j odd, j′ even: then −1 + r − s ≥ 0 and by (40)

\frac{∂^{2}Φ}{∂t^{2}}(t^{∗}, π/k) ≥ 2((1 + s)k^{2} − sl^{2} + s(1 + s)(l − k)^{2}) = 2(s(l − k) − k)^{2} ≥ 0;

j odd, j′ odd: then 1 − r − s ≥ 0. Considering (40), we have the following alternative. If l^{2} ≥ r(l − k)^{2}, then ∂^{2}Φ/∂t^{2}(t^{∗}, π/k) ≥ 0; otherwise

\frac{∂^{2}Φ}{∂t^{2}}(t^{∗}, π/k) ≥ 2(rk^{2} + (1 − r)(l^{2} − r(l − k)^{2})) = 2(r(l − k) − l)^{2} ≥ 0.

Let us show that then Φ^{∗} must decrease on [0, π/k]. Otherwise there are 0 ≤ ϑ_{1} < ϑ_{2} ≤ π/k such that Φ^{∗}(ϑ_{2}) > Φ^{∗}(ϑ_{1}). As π/k is a local minimum, there is a ϑ_{1} < ϑ^{∗} < π/k such that

Φ^{∗}(ϑ^{∗}) = max_{ϑ_{1}≤ϑ≤π/k} Φ^{∗}(ϑ) = max_{{0≤t<2π \atop ϑ_{1}≤ϑ≤π/k}} Φ(t, ϑ),

i. e. there further is some t^{∗} such that Φ has a local maximum in (t^{∗}, ϑ^{∗}). But then kt^{∗} ≡ lt^{∗} + ϑ^{∗} ≡ 0 mod. π and ϑ^{∗} ≡ 0 mod. π/k and this is impossible. That shows the proposition, except for one value of s at most for a given r. But Φ^{∗} is a continuous function of s and the proposition is true by a perturbation.

Example 10.1.3 The real and complex unconditionality constant of {0, 1, 2} in 𝒞(𝕋) is \sqrt{2}. Indeed, a case study shows that

∥1 + r e_{1} −s e_{2} ∥_{∞} = \{ \begin{matrix}r + |s − 1| & if r|s − 1| ≥ 4s \\ (1 + s)(1 + r^{2}/(4s))^{1/2} & if r|s − 1| ≤ 4s \end{matrix}

and this permits to compute the maximum (33), which is obtained for r = 2, s = 1.

Example 10.1.4 The real and complex unconditionality constant of {0, 1, 3} in 𝒞(𝕋) is 2/\sqrt{3}. Indeed, a case study shows that ∥1 + r e_{1} −s e_{3} ∥_{∞} makes

\{ \begin{matrix}1 + r − s & if s ≤ r/(4r + 9) \\ (\frac{2}{27}s(r^{2} + 9 + 3r/s)^{3/2} − \frac{2}{27}r^{3}s + \frac{2}{3}r^{2} + rs + s^{2} + 1)^{1/2} & if s ≥ r/(4r + 9) \end{matrix}

and this permits to compute the maximum (33), which is obtained for r = 3/2, s = 1/2.

These examples are particular cases of the following theorem.

Theorem 10.1.5 Let n_{1}, n_{2}, n_{3} ∈ 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 constant of E = {n_{1}, n_{2}, n_{3}} is

(cos \frac{π}{2n})^{−1}.

Proof. We may suppose n_{1} < n_{3} < n_{2}. Let k = n_{2} −n_{1} and l = n_{3} −n_{1}. By Lemma 10.1.2, the Arithmetic-Geometric Mean Inequality bounds the Sidon constant C of {n_{1}, n_{2}, n_{3}} in the following way:

C = max_{r,s>0} \frac{1 + r + s}{∥1 + r e_{k} + e^{iπ/k} s e_{l} ∥_{∞}} ≤ max_{r,s>0} \frac{1 + r + s}{|1 + r + e^{iπ/k} s|}
= max_{r,s>0}(1 − sin^{2} \frac{π}{2k} \frac{4s(1 + r)}{(1 + r + s)^{2}} )^{−1/2}
≤ (1 − sin^{2}(π/2k))^{−1/2} = (cos(π/2k))^{−1}.

This inequality is sharp: we have equality for r = l/(k − l) and s = 1 + r. In fact the derivative of |1 + r e^{ikt} + e^{iπ/k} s e^{ilt} |^{2} is then

\frac{8lk}{l − k} sin \frac{kt}{2} cos \frac{lt + π/k}{2} cos \frac{(l − k)t + π/k}{2}

and its extremal points are

\frac{2j}{k} π , \frac{2j + 1}{l}π − \frac{π}{lk} , \frac{2j + 1}{l − k}π − \frac{π}{(l − k)k} : j ∈ ℤ

so that its extremal values are

4s^{2} cos^{2} \frac{2j + 1}{2k}π , 4r^{2} sin^{2} \frac{2j + k + 1}{2l}π , 4 sin^{2} \frac{2j + k + 1}{2(l − k)}π : j ∈ ℤ.

Therefore the maximum of |1 + r e^{ikt} + e^{iπ/k} s e^{ilt} |^{2} is 4s^{2} cos^{2}(π/(2k)).

This proof and (34) yield also the more precise Proposition 10.1.6 The Sidon constant of {n_{1}, n_{2}, n_{3}} is attained for

ϑ_{1} |n_{2} − n_{3}| e_{n_{1}} +ϑ_{2} |n_{1} − n_{3}| e_{n_{2}} +ϑ_{3} |n_{1} − n_{2}| e_{n_{3}}

with ϑ_{1}, ϑ_{2}, ϑ_{3} ∈ {−1, 1} real signs such that ϑ_{1}ϑ_{2} = −1 if 2^{j} | n_{2} − n_{1} and 2^{j} ∤ n_{3} − n_{1} for some j; ϑ_{1}ϑ_{3} = −1 if 2^{j} ∤ n_{2} − n_{1} and 2^{j} | n_{3} − n_{1} for some j; ϑ_{2}ϑ_{3} = −1 otherwise. Thus complex and real unconditionality constants of E in 𝒞(𝕋) coincide.

Let us also underline the following easy consequences of our computation.

Corollary 10.1 (i) The Sidon constant of sets with three elements is at most \sqrt{2}. (ii) The Sidon constant of \{0, n, 2n\} is \sqrt{2}, while the Sidon constant of \{0, n+1, 2n\} is at most (cos(π/(2n))^{−1} = 1 + π^{2}/(8n^{2}) + o(n^{−2}).

10.2 Sidon constant of Hadamard sets

Recall that E = {n_{k}} ⊆ ℤ is a Hadamard set if there is a q > 1 such that n_{k+1}/n_{k} ≥ q for all k. It is a classical fact that then E is a Sidon set: Riesz products (see [62, Chapter 2]) even yield effective bounds for its Sidon constant. In particular, if q ≥ 3, then E’s Sidon constant is at most 2. Our computations provide an alternative proof for q > \sqrt{π^{2}/2 + 1} ≈ 2.44 and give a better bound for q > \sqrt{π^{2} + 1} ≈ 3.30. Putting k = 1 in (32) and using Theorem 10.1.5, we obtain

Corollary 10.2.1 Let E = {n_{k}} ⊆ ℤ.

(i) Let q > \sqrt{π^{2}/2 + 1}. If |n_{k+1}| ≥ q|n_{k}| for all k, then the Sidon constant of E is at most 1 + π^{2}/(2q^{2} − 2 − π^{2}). (ii) Let q ≥ 2 be an integer. If E ⊇ {n, n + k, n + qk} for some n and k, then the Sidon constant of E is at least (cos(π/2q))^{−1} ≥ 1 + π^{2}/(8q^{2}).

In particular, we have the following bounds for the Sidon constant C of G = {j^{k}}, j ∈ ℤ \ {−1, 0, 1}:

1 + π^{2}/(8(j + 1)^{2}) ≤ C ≤ 1 + π^{2}/(2j^{2} − 2 − π^{2}).

11 Density conditions

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 of E ⊆ ℤ is defined by

d^{∗}(E) = lim_{h→∞} max_{a∈ℤ} \frac{E ∩ \{a + 1, …, a + h\}}{h} .

Suppose E enjoys (ℐ_{n}) with n ≥ 2. Then E is a Λ(2n) set by Theorem 2.4.2(i). By [86, Th. 3.5] (see also [69, §1, Cor. 2]), d^{∗}(E) = 0. Now suppose E enjoys complex or real (𝒥_{n}) with n ≥ 2. As Li [58, Th. 2] shows, there are sets E such that 𝒞_{E}(𝕋) has ℓ_{1}-(map) while E contains arbitrarily long arithmetic sequences: we cannot apply Szemeredi’s Theorem. Kazhdan (see [40, Th. 3.1]) proved that if d^{∗}(E) > 1/n, then there is a t ∈ {1, …, n − 1} such that d^{∗}(E ∩ E + t) > 0. One might hope that it should in fact suffice to choose t in any interval of length n. However, Hindman [40, Th. 3.2] exhibits a counterexample: given s ∈ ℤ and positive ε, there is a set E with d^{∗}(E) > 1/2 − ε and there are arbitrarily large a such that E ∩ E − t = ∅ for all t ∈ {a + 1, …, a + s}. Thus, we have to be satisfied with

Lemma 11.1 Let E ⊆ ℤ with positive maximal density. Then there is a t ≥ 1 such that the following holds: for any s ∈ ℤ we have some a, |a| ≤ t, such that d^{∗}(E + a ∩ E + s) > 0.

Proof. By a result of Erdős (see [40, Th. 3.8]), there is a t ≥ 1 such that F = E + 1 ∪ … ∪ E + t satisfies d^{∗}(F) > 1/2. But then, by [40, Th. 3.4], d^{∗}(F ∩ F + s) > 0 for any s ∈ ℤ. This means that for any s there are 1 ≤ a, b ≤ t such that d^{∗}(E + a ∩ E + s + b) > 0.

We are now able to prove

Proposition 11.2 Let E ⊆ ℤ.

(i) If E has positive maximal density, then there is an a ∈ ℤ such that E ∪ {a} fails real (𝒥_{2}). Therefore E fails real (𝒥_{3}). (ii) If d^{∗}(E) > 1/2, then E fails real (𝒥_{2}).

Proof. (ii) is established in [58, Prop. 14]. (i) is a consequence of Lemma 11.1: indeed, if E has positive maximal density, then this lemma yields some a ∈ ℤ and an infinite F ⊆ E such that for all s ∈ F there are arbitrarily large k, l ∈ E such that k + a = l + s. Thus E ∪ {a} fails real (𝒥_{2}). Furthermore, E fails real (𝒥_{3}) by Proposition 8.1.2(iv).

Remark 11.3 We may reformulate the remaining open case of (𝒥_{2}). Let us introduce the infinite difference set of E: ΔE = {t : E ∩ (E − t) = ∞} (see [94] and [88]). Then E has real (𝒥_{2}) if and only if, for any a ∈ E, ΔE meets E − a finitely many times only. Thus our question is: are there sets with positive maximal density such that E − a ∩ ΔE is finite for all a ∈ E ?

Proposition 9.2.1 and Theorem 9.3.1 show that there is only one general condition of lacunarity on E that ensures properties (ℐ_{n}), (𝒥_{n}) or (ℐ_{∞}), (𝒥_{∞}): E must grow exponentially or superexponentially. 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} which nevertheless enjoy both (ℐ_{n}) and complex and real (𝒥_{n}). See [35, §II, (3.52)] for a proof in the case n = 2: it can be easily adapted to n ≥ 2 and shows also the way to construct, for any sequence n_{k} → ∞, sets that satisfy (ℐ_{∞}) and (𝒥_{∞}) and grow more slowly than k^{n_{k}}.

12 Unconditionality vs. probabilistic independence

12.1 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^{∞} be the Cantor group and Γ its dual group of Walsh functions. Consider the set R = {r_{i}} ⊆ Γ of Rademacher functions, i. e. of the coordinate functions on 𝔻^{∞}: they form a family of independent random variables that take values −1 and 1 with equal probability \frac{1}{2}: Thus ∥ ∑ ϵ_{i}a_{i}r_{i}∥_{X} does not depend on the choice of signs ϵ_{i} = ±1 for any homogeneous Banach space X on 𝔻^{∞} and R is a real 1-(ubs) in X. Clearly, R is also a complex (ubs) in all such X. But its complex unconditionality constant is π/2 [89] and L_{W}^{p} (𝔻^{∞}) has complex (umap) if and only if p = 2 or W = {w_{i}} ⊆ Γ is finite. Indeed, W would have an analogue property (𝒰) of block unconditionality in L^{p}(𝔻^{∞}): for any ε > 0 there would be n such that

max_{w∈𝕋} ∥ϵaw_{1} + bw_{n}∥_{p} ≤ (1 + ε)∥aw_{1} + bw_{n}∥_{p}.

But this is false: for 1 ≤ p < 2, take a = b = 1, ϵ = i:

max_{ϵ∈𝕋} ∥ϵw_{1} + w_{n}∥_{p} ≥ (\frac{1}{2}(|i + 1|^{p} + |i − 1|^{p}))^{1/p} = \sqrt{2} > ∥w_{1} + w_{n}∥_{p} = 2^{1−1/p};

for 2 < p ≤ ∞, take a = 1, b = i, ϵ = i:

max_{ϵ∈𝕋} ∥ϵw_{1} + iw_{n}∥_{p} ≥ (\frac{1}{2}(|i + i|^{p} + |i − i|^{p}))^{1/p} = 2^{1−1/p} > ∥w_{1} + iw_{n}∥_{p} = \sqrt{2}.

This is simply due to the fact that the image domain of the characters on 𝔻^{∞} is too small. Take now the infinite torus 𝕋^{∞} and consider the set S = {s_{i}} of Steinhaus functions, i. e. the coordinate functions on 𝕋^{∞}: they form again a family of independent random variables with values uniformly distributed in 𝕋. Then S is clearly a complex 1-(ubs) in any homogeneous Banach space X on 𝕋^{∞}.

12.2 Two notions of approximate probabilistic independence

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}}, …, e_{p_{n}}), p_{1}, …, p_{n} ∈ E, and to ask it to be close to the product of the distributions of the e_{p_{i}}. For example, Pisier [78, Lemma 2.7] gives the following characterization: E is a Sidon set if and only if there are a neighbourhood V of 1 in 𝕋 and 0 < ϱ < 1 such that for any finite F ⊆ E

m[e_{p} ∈ V : p ∈ F] ≤ ϱ^{|F |}.  (41)

Murai [72, §4.2] calls E ⊆ ℤ pseudo-independent if for all A_{1}, …, A_{n} ⊆ 𝕋

m[e_{p_{i}} ∈ A_{i} : 1 ≤ i ≤ n] ⟶_{{p_{i}∈E \atop p_{i}→∞}} ∏_{i=1}^{n} m[e_{p_{i}} ∈ A_{i}] = ∏_{i=1}^{n} m[A_{i}].  (42)

We have

Proposition 12.2.1 Let E ⊆ ℤ. The following are equivalent.

(i) E is pseudo-independent (ii) E enjoys (ℐ_{∞}). (iii) For every ε > 0 and m ≥ 1, there is a finite subset G ⊆ E such that the Sidon constant of any subset of E \ G with m elements is less than 1 + ε.

Note that by Corollary 2.4.3, (42) does not imply (41). Proof. (i) ⇔ (ii) follows by Proposition 8.1.1(iii) and [72, Lemma 30]. (iii) ⇒ (ii) is true because (iii) is just what is needed to draw our conclusion in Corollary 2.4.3. Let us prove (i) ⇒ (iii). Let ε > 0, m ≥ 1 and 𝒜 be a covering of 𝕋 with intervals of length ε. By (42), there is a finite set G ⊆ E such that for p_{1}, …, p_{m} ∈ E \ G and A_{i} ∈ 𝒜 we have m[e_{p_{i}} ∈ A_{i} : A_{i} ∈ 𝒜] > 0. But then

∥∑ a_{i} e_{p_{i}} ∥_{∞} ≥ ∑ |a_{i}| · (1 − ε).

Remark 12.2.2 (ii) ⇒ (iii) may be proved directly by the technique of Riesz products: see [45, Appendix V, §1.II].

Another possibility is to define some notion of almost independence. Berkes [2] introduces the following notion: let us call a sequence of random variables {X_{n}} almost i.i.d. (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}∥_{∞} → 0. We have the straightforward

Proposition 12.2.3 Let E = {n_{k}} ⊆ ℤ. If E is almost i.i.d., then E is a Sidon set with constant asymptotically 1.

Proof. Let {Y_{j}} be an i.i.d sequence and suppose ∥ e_{n_{j}} −Y_{j}∥_{∞} ≤ ε for j ≥ k. Then

∑_{j≥k} |a_{j}| = ∥∑_{j≥k} a_{j}Y_{j}∥_{∞} ≤ ∥∑_{j≥k} a_{j} e_{n_{j}} ∥_{∞} + ε ∑_{j≥k} |a_{j}|

and the unconditionality constant of {n_{k}, n_{k+1}, …} is less than (1 − ε)^{−1}.

Suppose E = {n_{k}} ⊆ ℤ is such that n_{k+1}/n_{k} is an integer for all k. In that case, Berkes [2] proves that E is almost i.i.d. if and only if n_{k+1}/n_{k} → ∞. We thus recover a part of Theorem 9.3.1. Question 12.2.4 What about the converse in Proposition 12.2.3 ?

13 Summary of results. Remarks and questions

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≥0} (j ∈ ℤ \ {−1, 0, 1}). The number given in the first (vs. second, third) column is the value n ≥ 1 for which the set in the corresponding row achieves exactly (ℐ_{n}) (vs. complex (𝒥_{n}), real (𝒥_{n})).

G = {j^{k}}_{k≥0} with |j| ≥ 2 (ℐ_{n}) ℂ-(𝒥_{n}) ℝ-(𝒥_{n})
G, j > 0 odd  |j|  |j|  ∞
G, j > 0 even  |j|  |j|  |j|
G ∪ {0}, j > 0 odd  |j|  |j| − 1  ∞
G ∪ {0}, j > 0 even  |j|  |j| − 1 |j| − 1
G, G ∪ {0}, j < 0 odd |j| − 1  |j|  ∞
G, G ∪ {0}, j < 0 even |j| − 1  |j|  |j|
G ∪ −G, G ∪ −G ∪ {0}, j odd  1  1  ∞
G ∪ −G, j even  1  1  |j|
G ∪ −G ∪ {0}, j even  1  1  |j|/2

Table 13.1

13.1 The case X = L^{p}(𝕋) with p an even integer

Let p ≥ 4 be an even integer. We observed the following facts.

Real and complex (umap) differ among subspaces L_{E}^{p}(𝕋) for each p: consider Proposition 9.1.1 or L_{E}^{p}(𝕋) with E = {±(p/2)^{k}}.

By Theorem 7.2.1, L_{E}^{p}(𝕋) has complex (vs. real) (umap) if so does L_{E}^{p+2}(𝕋);

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} ∪ {±p^{k}}.

Property (umap) is not stable under unions with an element: for each p, there is a set E such that L_{E}^{p}(𝕋) has complex (vs. real) (umap), but L_{E∪{0}}^{p}(𝕋) does not. In the complex case, consider E = {(p/2)^{k}}. In the real case, consider E = {±(2⌈p/4⌉)^{k}}.

If E is a symmetric set and p ≠ 2, then L_{E}^{p}(𝕋) fails complex (umap). Proposition 8.3.1 gives a criterion for real (umap).

What is the relationship between (umbs) and complex (umap) ? We have by Proposition 8.1.2(i) and 7.2.4(i)

Proposition 13.1.1 Let E = {n_{k}} ⊆ ℤ and n ≥ 1.

(i) If E is a (umbs) in L^{4n−2}(𝕋), then L_{E}^{2n}(𝕋) has complex (umap). (ii) If {π_{k}} realizes complex (umap) in L_{E}^{2n}(𝕋), then E is a (umbs) in L^{2n}(𝕋).

We also have, by Proposition 11.2(i)

Proposition 13.1.2 Let E ⊆ ℤ and p ≠ 2, 4 an even integer. If L_{E}^{p}(𝕋) has real (umap), then d^{∗}(E) = 0.

Note also this consequence of Propositions 3.3.1, 8.4.1, 12.2.1 and Theorems 2.4.2, 7.2.1

Proposition 13.1.3 Let σ > 1 and E = {[σ^{k}]}. Then the following properties are equivalent:

(i) σ is transcendental; (ii) L_{E}^{p}(𝕋) has complex (umap) for any even integer p; (iii) E is a (umbs) in any L^{p}(𝕋), p an even integer; (iv) E is pseudo-independent.

(v) For every ε > 0 and m ≥ 1, there is an l such that for k_{1}, …, k_{m} ≥ l the Sidon constant of {[σ^{k_{1}}], …, [σ^{k_{m}}]} is less than 1 + ε.

13.2 Cases X = L^{p}(𝕋) with p not an even integer and X = 𝒞(𝕋)

In this section, X denotes either L^{p}(𝕋), p not an even integer, or 𝒞(𝕋). Theorems 2.4.2 and 7.2.1 only permit us to use the negative results of Section 8: thus, we can just gather negative results about the functional properties of E. For example, we know by Proposition 8.1.2(iv) that (ℐ_{∞}) and (𝒥_{∞}) are stable under union with an element. Nevertheless, we cannot conclude that the same holds for (umap). The negative results are (by Section 8):

for any infinite E ⊆ ℤ, X_{E∪2E} fails real (umap). Thus (umap) is not stable under unions;

if E is a polynomial sequence (see Section 8), then E is not a (umbs) in X and X_{E} fails real (umap);

if E is a symmetric set, then E is not a (umbs) in X and X_{E} fails complex (umap). Proposition 8.3.1 gives a criterion for real (umap);

if E = {[σ^{k}]} with σ > 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).

Furthermore, by Proposition 9.1.1, real and complex (umap) differ in X. Theorem 9.3.1 is the only but general positive result on (umbs) and complex (umap) in X. Proposition 9.1.1 yields further examples for real (umap). What about the sets that satisfy (ℐ_{∞}) or (𝒥_{∞}) ? We only know that (ℐ_{∞}) does not even ensure Sidonicity by Corollary 2.4.3. One might wonder whether for some reasonable class of sets E, E is a finite union of sets that enjoy (ℐ_{∞}) or (𝒥_{∞}). This is false even for Sidon sets: for example, let E be the geometric sequence {j^{k}}_{k≥0} with j ∈ ℤ\{−1, 0, 1} and suppose E = E_{1}∪…∪E_{n}.

Then E_{i} = {j^{k}}_{k∈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| ≤ n. This means that there is an infinite subset B ⊆ A_{i} and an h, 1 ≤ h ≤ n, such that h + B ⊆ A_{i}. We may apply Proposition 8.1.2(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 13.1.1(ii) remain true for general X ? We do not know this. Suppose however that we know that {π_{k}} realizes (umap) in the following strong manner: for any ε > 0, a tail {π_{k}}_{k≥l} is a (1 + ε)-unconditional a.s. in X_{E}. Then E is trivially a (umbs) in X. In particular, this is the case if

1 + ε_{n} = sup_{ϵ∈𝕊} ∥Id − (1 + ϵ)π_{n}∥_{ℒ(X)}

converges so rapidly to 1 that ∑ ε_{n} < ∞. Indeed,

sup_{ϵ_{k}∈𝕊} ∥π_{n−1} + ∑_{k≥n} ϵ_{k}Δπ_{k}∥ ≤ (1 + ε_{n}) sup_{ϵ_{k}∈𝕊} ∥π_{n} + ∑_{k>n} ϵ_{k}Δπ_{k}∥.

and thus, for all f ∈ 𝒫_{E}(𝕋),

sup_{ϵ_{k}∈𝕊} ∥π_{l}f + ∑_{k>l} ϵ_{k}Δπ_{k}f∥ ≤ ∏_{k>l}(1 + ε_{k}) ∥f∥.

Let us finally state

Proposition 13.2.1 Let E ⊆ ℤ. If X_{E} has real (umap), then d^{∗}(E) = 0.

13.3 Questions

The following questions remain open:

Combinatorics Regarding Proposition 11.2(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 = ⋃ E_{i} in finite sets such that all E_{i} + E_{j}, i ≤ j, are pairwise disjoint ? Then L_{E}^{4}(𝕋) would admit a 1-unconditional (fdd) by Proposition 7.2.4(i).

Functional analysis Let X ∈ {L^{1}(𝕋), 𝒞(𝕋)} and consider Theorem 6.2.3. Is (𝒰) sufficient for X_{E} to share (umap) ? Is there a set E ⊆ ℤ such that some space L_{E}^{p}(𝕋), p not an even integer, has (umap), while 𝒞_{E}(𝕋) fails it ? It could be worthwile to look at certain subsets of E = {∑_{i∈F} n_{i} : F ⊂ ℕ finite} with a very rapidly growing sequence {n_{i}}. By [68, §VIII], it suffices to study E_{∞} = {∏_{i∈F} s_{i} : F ⊂ ℕ finite} in the dual group of 𝕋^{∞}. Then L_{E_{∞}}^{1}(𝕋^{∞}) has (uap) by an argument of Déchamps-Gondim. Does it have (umap) ? Does C_{E_{∞}}(𝕋^{∞}) enjoy the Daugavet property and thus fail (uap) ? Does the natural projection of C_{E_{∞}}(𝕋^{∞}) onto 𝒞(𝔻^{∞}) have a closed image ?

Harmonic analysis Is there a Sidon set E = {n_{k}} ⊆ ℤ of constant asymptotically 1 such that n_{k+1}/n_{k} is uniformly bounded ? What about the case E = [σ^{k}] for a transcendental σ > 1 ? If E enjoys (ℐ_{∞}), is E a (umbs) in L^{p}(𝕋) (1 ≤ p < ∞) ? What about (𝒥_{∞}) ?

Chapitre III

Unconditional entry basic sequences

1 Introduction

We study the following isometric question on the Schatten class S^{p}. How many matrix coefficients of an operator x ∈ 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 σ(p) sets of entries I ⊆ ℕ × ℕ, which have been introduced recently by Harcharras. We show that for our purpose, sets of matrix entries I ⊆ ℕ × ℕ are best understood as vertices for polygonal lines that follow alternatively the horizontal or the vertical axis in ℕ × 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. 3.4). The main step is a characterization of rectangular polygons that return on their track (Th. 2.7). 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 [36]. In the first place, we show that c.b. Λ(p) sets cannot contain the product of two infinite sets of characters. In the second place, we point out two new possibilities to construct Λ(p) sets.

Notation 𝕋 = {t ∈ ℂ : |t| = 1} is the unit circle endowed with its Haar measure dm and ℤ its dual group of integers: for each n ∈ ℤ, let e_{n}(t) = t^{n}. The cardinal of E ⊆ ℤ is written E . For a space X of integrable functions on 𝕋 and E ⊆ ℤ, X_{E} denotes the space of functions with Fourier spectrum in E: X_{E} = {f ∈ X : f̂(n) = ∫ e_{−n} f dm = 0 if n ∉ E}. The Schatten class S^{p}, 1 ≤ p < ∞, is the space of those compact operators x on ℓ_{2} such that ∥x∥_{p} = (tr |x|^{p})^{1/p} < ∞. The entry e_{rc} is the operator on ℓ_{2} that maps the cth basis vector on the rth basis vector. We shall also consider e_{rc} as the indicator function of (r, c) from ℕ × ℕ to ℕ. The matrix coefficient (r, c) of an operator x is x_{rc} = tr x e_{rc}^{∗} and its matrix representation is x ∼ ∑ x_{rc} e_{rc}. A Schur multiplier T on S^{p} associated to (μ_{rc})_{r,c≥0} is a bounded operator on S^{p} such that T e_{rc} = μ_{rc} e_{rc}. T is furthermore completely bounded (c.b. for short) if T ⊗ Id_{S^{p}} is bounded on S^{p}(ℓ_{2} ⊗_{2} ℓ_{2}) ≡ S^{p}(S^{p}). For I ⊆ ℕ × ℕ, the entry space S_{I}^{p} is the space of those x ∈ S^{p} whose matrix representation is supported by I: x_{rc} = 0 if (r, c) ∉ I. S_{I}^{p} is also the closed subspace of S^{p} spanned by {e_{q}}_{q∈I}.

2 Rectangular polygons in the two-dimensional integer lattice

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.

Definition 2.1 Let p = 2s ≥ 0 be an even integer and I ⊆ ℕ × ℕ.

(i) A rectangular line L in I is a sequence of vertices (q_{1}, …, q_{s}) ∈ 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. (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. (iv) A 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}), …, (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}) → (r_{1}, c_{2}) ↓ (r_{2}, c_{2}) → (r_{2}, c_{3}) ↓ … ↓ (r_{s}, c_{s}) → (r_{s}, c_{1}) ↓,

where the last arrow ↓ indicates the segment (r_{s}, c_{1}) ↓ (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.

(v) Let A_{s}^{I} = {α ∈ ℕ^{I} : ∑_{q∈I} α_{q} = s}. The set of p-gonal relations in I is

Γ_{s}^{I} = {(α, β) ∈ A_{s}^{I} × A_{s}^{I} : ∀r ∑_{c} α_{rc} = ∑_{c} β_{rc} & ∀c ∑_{r} α_{rc} = ∑_{r} β_{rc}}.

The set P_{s}^{I} of connected p-gonal relations in I is the subset of those (α, β) ∈ Γ_{s}^{I} that cannot be decomposed into two other nonempty polygonal relations without row nor column in common:

\{ \begin{matrix}(α, β) = (α^{1}, β^{1}) + (α^{2}, β^{2}) \\ with (α^{i}, β^{i}) ∈ Γ_{s_{i}}^{I} & s_{i} ≥ 1 \end{matrix} ⟹  | \begin{matrix}∃r, c_{1}, c_{2} α_{rc_{1}}^{1}, α_{rc_{2}}^{2} ≥ 1 or \\ ∃c, r_{1}, r_{2} α_{r_{1}c}^{1}, α_{r_{2}c}^{2} ≥ 1 \end{matrix}

Example 2.2 (i) for an octagon in {0, 1} × {0, 1} ∪ {1, 2} × {2, 3}:

(0, 0) → (0, 1) ↓ (1, 1) → (1, 3) ↓ (2, 3) → (2, 2) ↓ (1, 2) → (1, 0) ↓ .

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

The next proposition shows that, for our purpose, connected p-gonal relations describe completely p-gons.

2. rectangular polygons in the two-dimensional integer lattice 73

Proposition 2.3 Let p = 2s ≥ 0 be an even integer and I ⊆ ℕ × ℕ. The mapping P = ((r_{1}, c_{1}), (r_{1}, c_{2}), …, (r_{s}, c_{1})) ↦ (α, β), where

α_{q} = {i : (r_{i}, c_{i}) = q} & β_{q} = {i : (r_{i}, c_{i+1}) = q} ,

is a surjection of the set of p-gons in I onto the set P_{s}^{I} of connected p-gonal relations in I. We shall write P ∼ (α, β) and call γ_{αβ} the number of p-gons mapped on (α, β).

Proof. Let (α, β) ∈ P_{s}^{I}. Consider a rectangular line

(r_{1}, c_{1}) → (r_{1}, c_{2}) ↓ … → (r_{j}, c_{j+1}) ∈ I^{2j}

such that α_{q}^{1} = {i : (r_{i}, c_{i}) = q} ≤ α_{q} and β_{q}^{1} = {i : (r_{i}, c_{i+1}) = q} ≤ β_{q} and j is maximal. We claim (a) that c_{j+1} = c_{1} and (b) that j = s. Let (α^{2}, β^{2}) = (α, β) − (α^{1}, β^{1}).

(a) If c_{j+1} ≠ c_{1}, then ∑_{r} α_{rc_{j+1}}^{2} = ∑_{r} β_{rc_{j+1}}^{2} + 1 ≥ 1. Thus there is r_{j+1} such that α_{r_{j+1}c_{j+1}}^{2} ≥ 1. But then ∑_{c} β_{r_{j+1}c}^{2} = ∑_{c} α_{r_{j+1}c}^{2} ≥ 1 and there is c_{j+2} such that β_{r_{j+1}c_{j+2}}^{2} ≥ 1: j is not maximal.

(b) Suppose j < s. Then (α^{2}, β^{2}) ∈ Γ_{s−j}^{I} is nonempty. As (α, β) is connected, there is r, c, c′ such that α_{rc}^{1}, α_{rc′}^{2} ≥ 1 or r, r′, c such that α_{rc}^{1}, α_{r′c}^{2} ≥ 1. Now our problem is invariant under transposition; further it is invariant under the cyclic permutations of even order

(r_{k}, c_{k}) → (r_{k}, c_{k+1}) ↓ … → (r_{k−1}, r_{k}) ↓ .  (1)

We may thus conclude without loss of generality that for r′_{1} = r_{j} there is c′_{1} such that α_{r′_{1}c′_{1}}^{2} ≥ 1. Then there is c′_{2} such that β_{r′_{1}c′_{2}}^{2} ≥ 1. By the argument of (a), there is a 2j′-gon in (α^{2}, β^{2}) of the form (r′_{1}, c′_{1}) → (r′_{1}, c′_{2}) ↓ … → (r′_{j′}, c′_{1}) ↓. Then the following 2(j + j′)-gon

(r_{1}, c_{1}) → … ↓ (r_{j}, c_{j}) → (r′_{1}, c′_{2}) ↓ … → (r′_{j′}, c′_{1}) ↓ (r′_{1}, c′_{1}) → (r_{j}, c_{1}) ↓

shows that j is not maximal.

We now introduce combinatorial properties in order to visualize the special class of p-gons that we shall discover in the next section.

Definition 2.4 Let L be a closed rectangular line q_{1} → q_{2} → … → q_{p} → q_{1}.

(i) L does not span a surface if the index of every point in ℝ × ℝ is 0 with respect to L: the bounded open set inside L is empty. (ii) L returns on its track if L takes every elementary segment [(r, c), (r, c + 1)] and [(r, c), (r + 1, c)] in ℕ × ℕ as many times in one sense as in the other sense. (iii) Two rectangular lines are similar if they are equivalent up to juxtaposed rectangular lines that return on their track.

Proposition 2.5 Let P ∼ (α, β) be the 2s-gon (r_{1}, c_{1}) → … → (r_{s}, c_{1}) ↓ and q ∈ ℕ × ℕ.

(i) A closed rectangular line is similar to a polygon. (ii) If α_{q}, β_{q} ≥ 1, 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} ∼ (α^{1}, β^{1}), P_{2} ∼ (α^{2}, β^{2}) such that (α, β) = (α^{1}, β^{1}) + (α^{2}, β^{2}) + (e_{q}, e_{q}). (iii) If α_{q} ≥ 2 (vs. β_{q} ≥ 2), then P is the juxtaposition of two nonempty polygons P_{1} ∼ (α^{1}, β^{1}), P_{2} ∼ (α^{2}, β^{2}) such that α_{q}^{1}, α_{q}^{2} ≥ 1 (vs. β_{q}^{1}, β_{q}^{2} ≥ 1).

Proof. (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 (1). Let 1 ≤ j ≤ s be such that (r_{j}, c_{j+1}) = q. Consider the two following polygons:

P_{1} = (r_{j}, c_{j}) → (r_{1}, c_{2}) ↓ … → (r_{j−1}, c_{j}) ↓,
P_{2} = (r_{j+1}, c_{j+1}) → … ↓ (r_{s}, c_{s}) → (r_{s}, c_{1}) ↓ .

P is the juxtaposition of (r_{1}, c_{1}) → (r_{1}, c_{2}), P_{1} deprived of its first segment, (r_{j}, c_{j}) → (r_{j}, c_{j+1}) ↓ (r_{j+1}, c_{j+1}), P_{2} deprived of its last segment and (r_{s}, c_{1}) ↓ (r_{1}, c_{1}). Thus P is equivalent to the juxtaposition of P_{1} and P_{2}, with in between (r_{1}, c_{1}) → (r_{1}, c) → (r_{1}, c_{1}), where c is the point among c_{1}, c_{2}, c_{j} which is between the two others plus (r_{1}, c_{1}) ↓ (r, c_{1}) ↓ (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 2.3(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 (1), we may suppose without loss of generality that α_{q} ≥ 2 and q = (r_{1}, c_{1}). Let j be such that (r_{j}, c_{j}) = q. Set

P_{1} = (r_{1}, c_{1}) → … → (r_{j−1}, c_{j}) ↓ , P_{2} = (r_{j}, c_{j}) → … → (r_{s}, c_{1}) ↓ .

Corollary 2.6 Let P ∼ (α, β) be a rectangular polygon.

(i) P is equivalent to the juxtaposition of closed rectangular lines that return on their track and disconnected polygons P_{j} ∼ (α^{j}, β^{j}) such that α_{q}^{j} = 0 or β_{q}^{j} = 0 for every q ∈ ℕ × ℕ. (ii) If α = β, then P is equivalent to the juxtaposition of connected closed rectangular lines that return on their track: P is similar to the empty polygon. (iii) P is the juxtaposition of polygons P_{j} ∼ (α^{j}, β^{j}) such that α_{q}^{j}, β_{q}^{j} ≤ 1 for every q ∈ ℕ × ℕ.

Proof. (i) Use Proposition 2.5(ii) in a maximality argument and note that closed curves contained in disconnected polygons are disconnected. (ii) Note that if α = β, then α^{j} = β^{j} in (i): the P_{j}’s are disconnected. (iii) Apply Proposition 2.5(iii) in a maximality argument.

We are now able to describe precisely the combinatorial properties introduced in Definition 2.4.

Theorem 2.7 Let P ∼ (α, β) be the 2s-gon (r_{1}, c_{1}) → … → (r_{s}, c_{1}) ↓. The following are equivalent.

(i) α = β. (ii) P returns on its track. (iii) P does not span a surface. (iv) For every vertex q of P, there are as many row segments in P reaching q as leaving q.

(v) For every vertex q of P, there are as many column segments in P reaching q as leaving q.

Proof. (i) ⇒ (ii) is Corollary 2.6(ii). (ii) ⇒ (iii) follows from the definition of the index as a path integral. (iii) ⇒ (iv) Let q ∈ ℕ×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 [15, VII(6.6)]. 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 α_{q} = β_{q}. (iv) ⇒ (v) Because P is a closed curve, there are as many segments reaching q as leaving q.

(v) ⇒ (i) Let q ∈ ℕ × ℕ. Note that α_{q} is exactly the number of column segments reaching q plus {j : (r_{j−1}, c_{j}) = (r_{j}, c_{j}) = q}; β_{q} is exactly the number of column segments leaving q plus {j : (r_{j}, c_{j+1}) = (r_{j+1}, c_{j+1}) = q}.

3 1-unconditional basic sequences of entries

Let us recall the following definitions.

Definition 3.1 ([36, §4]) Let I ⊆ ℕ × ℕ and p ≥ 1.

(i) I is an unconditional basic sequence in S^{p} if for some C

sup_{±}∥∑_{q∈I} ±a_{q} e_{q}∥_{p} ≤ C∥∑_{q∈I} a_{q} e_{q}∥_{p}.

This amounts to the uniform boundedness of the family of relative Schur multipliers by signs

M_{ϵ}: S_{I}^{p} → S_{I}^{p}, (x_{rc}) ↦ (ϵ_{rc}x_{rc}) with |ϵ_{rc}| = 1.  (2)

I is complex (vs. real) 1-unconditional if all these multipliers are isometries for ϵ_{rc} ∈ 𝕋 (vs. ϵ_{rc} ∈ 𝔻). Then the norm of x ∈ S_{I}^{p} does not depend on the complex (vs. real) signs of its matrix coefficients. (ii) I is a c.b. unconditional basic sequence if the family (2) is furthermore uniformly c.b. I is a complex (vs. real) c.b. 1-unconditional basic sequence if the family (2) consists of c.b. isometries for ϵ_{rc} ∈ 𝕋 (vs. ϵ_{rc} ∈ 𝔻).

Notorious examples are single rows, single columns, single diagonals, single antidiagonals and more generally sets I such that for each (r, c) ∈ I, there is no other element of I in the row r or in the column c.

We now undertake the matrix counterpart of the computation presented in Section II.2.2

Computational lemma 3.2 Let p = 2s be an even positive integer and I ⊆ ℕ×ℕ. Put

Φ_{I}(ϵ, z) = tr |∑_{q∈I} ϵ_{q}z_{q} e_{q}|^{p} for ϵ ∈ 𝕊^{I} and z ∈ ℂ^{(I)}.

Then

Φ_{I}(ϵ, z) = ∑_{(α,β)∈P_{s}^{I}} γ_{αβ}ϵ^{β−α}\overline{z}^{α}z^{β}  (3)

where γ_{αβ} is a positive integer for every (α, β) ∈ P_{s}^{I}.

Proof. Let us expand Φ_{I}.

Φ_{I}(ϵ, z) = tr( ∑_{(r,c),(r′,c′)∈I} ϵ_{rc}^{−1}\overline{z_{rc}} e_{cr} ϵ_{r′c′}z_{r′c′} e_{r′c′})^{s}
= tr ∑_{{(r_{1},c_{1}),(r′_{1},c′_{1}),..., \atop (r_{s},c_{s}),(r′_{s},c′_{s})∈I}} ∏_{i=1}^{s} ϵ_{r_{i}c_{i}}^{−1} \overline{z_{r_{i}c_{i}}} e_{c_{i}r_{i}} ϵ_{r′_{i}c′_{i}}z_{r′_{i}c′_{i}} e_{r′_{i}c′_{i}}

= ∑_{{(r_{1},c_{1}),(r_{1},c_{2}),..., \atop (r_{s},c_{s}),(r_{s},c_{1})∈I}} ∏_{i=1}^{s} ϵ_{r_{i}c_{i}}^{−1}ϵ_{r_{i}c_{i+1}} \overline{z_{r_{i}c_{i}}}z_{r_{i}c_{i+1}}  (4)

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 (α, β), Proposition 2.3 yields (3).

The following definition shows up in the analysis of the above computation.

Definition 3.3 Let I ⊆ ℕ × ℕ and s ≥ 1. I is matrix s-independent if, given q, q′ ∈ 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.

Computational lemma 3.2 and Theorem 2.7 yield now the main theorem of this chapter.

Theorem 3.4 Let I ⊆ ℕ × ℕ and p = 2s a positive even integer. The following assertions are equivalent.

(i) I is a c.b. complex 1-unconditional basic sequence in S^{p}. (ii) I is a complex 1-unconditional basic sequence in S^{p}. (iii) I is a real 1-unconditional basic sequence in S^{p}. (iv) every 2s′-gon P in I with s′ ≤ s satisfies the equivalent properties (i)–(v) in Theorem 2.7.

(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. (vi) I is matrix s-independent.

Proof. (i) ⇒ (ii) ⇒ (iii) is trivial. (iii) ⇒ (iv) If I is a real 1-(ubs) and P ∼ (α, β) is a 2s-gon in I, then α ≡ β mod. 2 by Computational lemma 3.2. Take now a 2s′-gon P ∼ (α, β) in I. Suppose first that α_{q}, β_{q} ≤ 1 for all q ∈ I. Let q_{1} be the first vertex of P and P′ be the juxtaposition of the 2(s − s′)-gon q_{1} → q_{1} ↓ … → q_{1} on P. Then P′ ∼ (α, β) + (s − s′)(e_{q}, e_{q}) and, by hypothesis, α + (s − s′) e_{q} ≡ β + (s − s′) e_{q} mod. 2 and α = β. Suppose now that P is a general 2s′-gon P ∼ (α, β) in I. Then, by Corollary 2.6(iii), P is the juxtaposition of polygons P_{j} ∼ (α^{j}, β^{j}) such that α_{q}^{j}, β_{q}^{j} ≤ 1 for every q ∈ ℕ × ℕ. But then α^{j} = β^{j} for each j and thus α = β. (iv) ⇒ (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′. The juxtaposition of L, the column segment from the last point of L to the last point of L′, −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) ⇒ (vi) is trivial. (vi) ⇒ (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) ⇒ (i) holds by Computational lemma 3.2 and Theorem 2.7: 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 (α, β) ∈ P_{s}^{I} satisfies α = β, then (4) is constant in ϵ.

Remark 3.5 Harcharras [36] used Peller’s discovery [76] of the link between Fourier and Hankel Schur multipliers to produce unconditional basic sequences in S^{p} of the type E^{⌢} = {(r, c) ∈ ℕ × ℕ : r + c ∈ E}, where E ⊆ ℤ. Such sets E^{⌢} are matrix 2-independent if and only if E is 2-independent in the sense of Section II.2.2. But there are 3-independent sets E such that E^{⌢} is not matrix 3-independent: consider

(n_{1}, 0) → (n_{1}, n_{2}) ↓ (0, n_{2}) → (0, n_{1}) ↓ (n_{2}, n_{1}) → (n_{2}, 0) ↓

with n_{3} = n_{1} + n_{2}, n_{2} > n_{1} > 0 and 4n_{2} ≠ 3n_{3}.

4 Two remarks on a paper by Harcharras

4.1 The c.b. unconditionality constant of sum sets

We generalize Harcharras’ [36, Prop. 2.8].

Proposition 4.1 Let A, B ⊆ ℤ with A = B = n. Then, for any p ≥ 1, the c.b. unconditionality constant of A + B in L^{p}(𝕋) is at least ⌊n^{1/3}⌋^{|1/2−1/p|}.

Proof. We shall use an inductive construction of sets A_{i} ⊆ A and B_{i} ⊆ B such that A_{i} = B_{i} = i and

∀a, a′ ∈ A_{i} ∀b, b′ ∈ B_{i} a + b = a′ + b′ ⇒ a = a′ and b = b′.  (5)

Put a_{1} = min A and b_{1} = min B. Assume A_{i} and B_{i} are constructed. Put

a_{i+1} = min{a ∈ A \ A_{i} : ∀a′ ∈ A_{i} ∀b ≠ b′ ∈ B_{i} a + b ≠ a′ + b′}

if such an element exists; else A \ A_{i} ⊆ A_{i} + (B_{i} − B_{i}) \ {0} and n − i ≤ i^{2}(i − 1). If we are able to construct A_{i+1} = A_{i} ∪ {a_{i+1}}, put

b_{i+1} = min{b ∈ B \ B_{i} : ∀b′ ∈ B_{i} ∀a ≠ a′ ∈ A_{i+1} a + b ≠ a′ + b′}

if such an element exists; else B \B_{i} ⊆ B_{i} +(A_{i+1} −A_{i+1})\{0} and n−i ≤ i^{2}(i+1). We conclude from this construction that there are A_{i} = {a_{1}, …, a_{i}} ⊆ A and B_{i} = {b_{1}, …, b_{i}} ⊆ B of cardinality the minimal i such that n ≤ i^{3} + i^{2} + i, and a fortiori with i = ⌊n^{1/3}⌋. The end of the proof is the same as Harcharras’. The unconditionality constant of the canonical basis of S_{p}^{i} is i^{|1/2−1/p|}: there is a Schur multiplier M : S_{p}^{i} → S_{p}^{i}, (x_{kl}) ↦ (ϵ_{kl}x_{kl}), such that |ϵ_{kl}| = 1 and ∥M∥ = i^{|1/2−1/p|}. Put

f_{x} = (e_{a_{k}+b_{l}} x_{kl}) = [ \begin{matrix}e_{a_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & e_{a_{i}} \end{matrix} ] x [ \begin{matrix}e_{b_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & e_{b_{i}} \end{matrix} ] ∈ L_{A_{i}+B_{i}}^{p}(S_{p}^{i}).

Then ∥f_{x}(t)∥_{S_{p}^{i}} = ∥x∥_{S_{p}^{i}} for each t ∈ 𝕋 and thus ∥f_{x}∥_{L^{p}(S_{p}^{i})} = ∥x∥_{S_{p}^{i}}. Consider ν : A_{i} + B_{i} → 𝕋 be such that ν(a_{k} + b_{l}) = ϵ_{kl}. By (5), ν is well defined. Let N be the operator of convolution with ∑_{j∈A_{i}+B_{i}} ν(j) e_{j} acting on L_{A_{i}+B_{i}}^{p}(𝕋). As

N ⊗ Id_{S_{p}^{i}}(f_{x}) = (e_{a_{k}+b_{l}} ϵ_{kl}x_{kl}) = f_{Mx},

∥N ⊗ Id_{S_{p}^{i}}(f_{x})∥ = ∥Mx∥. Now the c.b. unconditionality constant of A_{i} + B_{i} is at least ∥N ⊗ Id_{S_{p}^{i}}∥ ≥ ∥M∥.

Corollary 4.2 If E ⊆ ℤ contains the sum of two infinite sets, then E is not a c.b. Λ(p) set for any p > 2.

Example 4.3 E = {2^{i} − 2^{j} : i > j} is not a c.b. Λ(p) set for any p > 2. Indeed, {2^{i} − 2^{j}} = E ∪ −E does not and if E did, then also −E and E ∪ −E.

4.2 Two new sufficient conditions for Λ(2s) sets

The proof of [36, Th. 1.14] and especially [36, Prop. 1.14] contain implicitly the two following new means to construct Λ(p) sets.

Proposition 4.4 Let E ⊆ ℤ and s ≥ 2 and integer. Let

r′_{s}(E, n) = {q_{1} < … < q_{s} : q_{1} + q_{2} + … + q_{s} = n};
z_{s}(E, n) = {q ∈ E^{s} : q_{i} distinct & − q_{1} + q_{2} − … + (−1)^{s}q_{s} = n} .

E is a Λ(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.

The number r′_{s} simplifies Rudin’s [86, 1.6(b)] number r_{s} in that it considers only distinct q_{i}. This is very useful in applications. The number z_{s} is [36, Def. 1.11]; Harcharras proves that if z_{s}(E, n) is a bounded function of n, then E is even a c.b. Λ(p) set. Nevertheless we wish to point out that the condition is new even for usual Λ(p) sets.

Chapitre IV

Random constructions inside lacunary sets

1 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 [59] uses in his argumentation a construction due to Katznelson [51] to discriminate the following two functional properties of certain subsets E ⊆ ℤ:

A Lebesgue integrable function on the circle with Fourier frequencies in E is in fact p-integrable for all p < ∞. This means that all spaces L_{E}^{p}(𝕋) coincide for p < ∞, i. e. E is a Λ(p) set for all p in Rudin’s terminology. No sequence of polynomial growth has this property [86, Th. 3.5]. By Theorem 4.7, almost every sequence of a given superpolynomial order of growth is Λ(p) for all p.

A bounded measurable function on the circle with Fourier frequencies in E is in fact continuous up to a set of measure 0. This means that L_{E}^{∞}(𝕋) and 𝒞_{E}(𝕋) coincide: E is a Rosenthal set. Every sequence of exponential growth is a Sidon set and therefore has this property. By Bourgain’s Theorem 2.5, almost every sequence of a given subexponential order of growth fails the Rosenthal property.

A Rosenthal set may contain arbitrarily large intervals [84] und thus fail the Λ(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 Λ(p) for all p and fails Rosenthal: his construction witnesses for the quantitative overlap between superpolynomial and subexponential order of growth.

We come back to Li [59] 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 4. In the second place, we want to precise and supple the random construction in the following sense: can one distinguish the Λ(p) property and the Rosenthal property among subsets of a certain given set ? That sort of questions has been investigated by Bourgain in [9]. We give the following answer (see Th. 2.8):

Main Theorem Consider a polynomial sequence of integers, or the sequence of primes. Then some subsequence of it is Λ(p) for all p and at the same time fails the Rosenthal property.

This is a special case of the more general question: does every set that fails the Rosenthal property contain a subset that is Λ(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 [30] 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 E ⊆ ℤ.

Any integer n has at most one representation as the sum of s elements of E. This implies that E is Λ(2s) by [86, Th. 4.5(b)].

E is equidistributed in Hermann Weyl’s sense: save for t ≡ 0 mod 2π, the successive means of {e^{int}}_{n∈E} tend to 0, which is the mean of e^{it} over [0, 2π[. This implies that E is not Rosenthal by [64, Lemma 4].

Our random construction gives no hint for explicit procedures to build such integer sets. The question whether some “natural” set of integers is Λ(p) for all p and fails the Rosenthal property remains open. Let us describe the paper briefly. Section 2 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 3, and show that they are nicely distributed among the intervals of the partition of ℤ defined by {±2^{k!}}. Section 4 establishes an optimal criterion for the generic subset of a set with polynomial growth to be Λ(p) for all p. Section 5 comes back to Bourgain’s proof in [8, Prop. 8.2(i)]: we simplify and strengthen it in order to investigate the generic subset of an equidistributed set.

Notation 𝕋 = {t ∈ ℂ : |t| = 1} is the unit circle endowed with its Haar measure dm and ℤ its dual group of integers: for each n ∈ ℤ, let e_{n}(t) = t^{n}. The cardinal of E = {n_{k}} ⊆ ℤ is written E . 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 E ⊆ ℤ, X_{E} denotes the space of functions with Fourier spectrum in E: X_{E} = {f ∈ X : f̂(n) = ∫ e_{−n} f dm = 0 if n ∉ E}. We shall stick to Hardy’s notation: u_{n} ≼ v_{n} (vs. u_{n} ≪ v_{n}) if u_{n}/v_{n} is bounded (vs. vanishes) at infinity.

2. equidistributed and Λ(p) sets 81

2 Equidistributed and Λ(p) sets

Definition 2.1 Let E = {n_{k}}_{k≥1} ⊆ ℤ ordered by increasing absolute value |n_{k}|.

(i) [86, Def. 1.5] Let p > 0. E is a Λ(p) set if, for some — or equivalently for any — 0 < r < p, L_{E}^{p}(𝕋) and L_{E}^{r}(𝕋) coincide:

∃C_{r} ∀f ∈ L_{E}^{p}(𝕋) ∥f∥_{r} ≤ ∥f∥_{p} ≤ C_{r}∥f∥_{r}.

(ii) [98, §7] E is equidistributed if for each t ∈ 𝕋 \ {1} the successive means

f_{k}(t) = \frac{1}{k} ∑_{j=1}^{k} e_{n_{j}}(t) ⟶_{k→∞} 0.  (1)

Thus E is equidistributed if and only if the sequence of characters in E converges to 1_{{1}} for the Cesàro summing method. If f_{k} tends pointwise to f ∈ c_{0}(𝕋), then E is weakly equidistributed.

If E is weakly equidistributed, then f defines an element of 𝒞_{E}(𝕋)^{⊥⊥}. By Lust-Piquard’s [64, Lemma 4], 𝒞_{E}(𝕋) then contains a copy of c_{0} and E cannot be Rosenthal. For example, ℤ and ℕ are equidistributed. Arithmetic sequences are weakly equidistributed: there is a finite set on which f_{k} ↛ 0. Polynomial sequences of integers ([98, Th. 9] and [96, Lemma 2.4], see [65, Ex. 2]) and the sequence of prime numbers (Vinogradov’s theorem [20], see [65, Ex. 1]) are weakly equidistributed: f_{k}(t) may not converge to 0 for rational t. There are nevertheless sequences of bounded pace that are not weakly equidistributed [23, Th. 11]. Sidon sets are Λ(p) for all p [86, Th. 3.1], but not weakly equidistributed since they are Rosenthal.

Example 2.2 Consider the geometric sequence E = {3^{k}}_{k≥1} and the corresponding sequence of successive means f_{k}. By [23, Th. 14], the f_{k} do not converge to 0 on a null set of Hausdorff dimension 1. Consider

f_{k}^{j} = k^{−j} ∑_{1≤k_{1},...,k_{j}≤k} e_{3^{k_{1}} +...+3^{k_{j}}} = k^{−j}( j! ∑_{1≤k_{1}<...<k_{j}≤k} + ∑_{1≤k_{1},...,k_{j}≤k not all distinct}) e_{3^{k_{1}} +...+3^{k_{j}}} .

Let j ≥ 1. Put E^{(j)} = {3^{k_{1}} + … + 3^{k_{j}} : 1 ≤ k_{1} < … < k_{j}} and let f_{k}^{(j)} be the corresponding successive means (1). Then

∥f_{k}^{j} − f_{({k \atop j})}^{(j)}∥_{∞} ≤ (({k \atop j} )^{−1} − \frac{j!}{k^{j}} )({k \atop j} ) + \frac{1}{k^{j}} (k^{j} − \frac{k!}{(k − j)!})
= 2(1 − \frac{k!}{k^{j}(k − j)!}) ⟶_{k} 0.

Thus E^{(j)}, which is Λ(p) for all p [67, Th. IV.3] and not Sidon, is not weakly equidistributed.

However, as Li notes, these two classes meet.

Theorem 2.3 ([59]) There is an equidistributed sequence that is Λ(p) for all p.

Sketch of proof. Li uses the following random construction, discovered by Erdős [21, 22] and introduced to harmonic analysis by Katznelson and Malliavin [52, 53].

Construction 2.4 Let E ⊆ ℤ and consider independent {0, 1}-valued selectors ξ_{n} of mean δ_{n} (n ∈ E), i. e. ℙ [ξ_{n} = 1] = δ_{n}. Then the random set E′ is defined by

E′ = {n ∈ E : ξ_{n} = 1}.

The first ingredient of the proof is Bourgain’s following

Theorem 2.5 ([8, Prop. 8.2(i)]) Let E = ℕ in Construction 2.4. If δ_{n} decreases with n while δ_{n} ≫ n^{−1}, then E′ is almost surely equidistributed.

Remark 2.6 In this sense, almost every sequence of a subexponential growth given by {δ_{n}} is equidistributed: indeed, for almost every E′ ⊆ ℕ,

E′ ∩ [0, n] ∼ δ_{0} + … + δ_{n} ≫ log n

by the Law of Large Numbers. Note however that the set E^{(j)} defined in Example 2.2 has subexponential growth: E^{(j)} ∩ [−n, n] ≽ (log n)^{j}, and is not equidistributed.

The second ingredient is a result announced without proof by Katznelson.

Proposition 2.7 ([51, §2]) Put I_{k} = ]p_{k−1}, p_{k}] with p_{k} > p_{k−1}^{2} (k ≥ 1). Let E = ℕ in Construction 2.4. There is a choice of (ℓ_{k}) with ℓ_{k} ≫ log p_{k} such that for δ_{n} = ℓ_{k}/ I_{k} (n ∈ I_{k}), E′ is Λ(p) for all p almost surely.

Li suggests to apply the content of Proposition 2.7 with p_{k} = 2^{k} and ℓ_{k} = k: then δ_{n} ≫ n^{−1} and Theorem 2.3 derives from Theorem 2.5.

We shall generalize Katznelson’s and Li’s results with a new proof that permits to construct E′ inside of sets E with polynomial growth (see Def. 3.1) and yields an optimal criterion on ℓ_{k}. We shall subsequently generalize Theorem 2.5 to obtain the Main Theorem via

Theorem 2.8 Let E be equidistributed (vs. weakly) and with polynomial growth. Then there is a subset E′ ⊆ E equidistributed (vs. weakly) and at the same time Λ(p) for all p.

See also Corollary 5.5 for a precise and quantitative statement.

3 Sets with polynomial growth

We start with the definition and first property of such sets.

Definition 3.1 Let E = {n_{k}}_{k≥1} ⊆ ℤ be an infinite set ordered by increasing absolute value and E[t] = E ∩ [−t, t] its distribution function.

(i) E has polynomial growth if n_{k} ≼ k^{d} for some 1 ≤ d < ∞. This amounts to E[t] ≽ t^{ε} for ε = d^{−1}. (ii) E has regular polynomial growth if there is a c > 1 such that |n_{⌈ck⌉}| ≤ 2|n_{k}| for large k. This amounts to E[2t] ≥ cE[t] for large t.

4. sets that are Λ(p) for all p 83

Proof. (i) If |n_{k}| ≤ Ck^{d} for large k and Ck^{d} ≤ t < C(k + 1)^{d}, then E[t] ≥ k > (t/C)^{ε} − 1. Conversely, if E[t] ≥ ct^{ε} for large t and c(t − 1)^{ε} < k ≤ ct^{ε}, then |n_{k}| ≤ t < (k/c)^{d} + 1. (ii) If |n_{⌈ck⌉}| ≤ 2|n_{k}| for large k and k is maximal with |n_{k}| ≤ t, then E[2t] ≥ E[2|n_{k}|] ≥ ck = cE[t]. Conversely, if E[2t] ≥ cE[t] for large t, then E[|n_{k}|] ∈ {k, k + 1} and E[2|n_{k}|] ≥ ck. Thus |n_{⌈ck⌉}| ≤ 2|n_{k}|.

In particular, polynomial sequences have regular polynomial growth. By the Prime Number Theorem, the sequence of primes also has. Property (ii) implies property (i): if E[2t] ≥ cE[t] for large t, then E[t] ≽ t^{log_{2} c}. The converse however is false as shows F = ⋃]2^{2^{2k}}, 2^{2^{2k}+1}], for which F [t] ≽ t^{1/4} while F [2t] = F [t] infinitely often. Let us relate Definition 3.1 with certain partitions of ℤ. Regular growth means in fact that E is regularly distributed on the annular dyadic partition of ℤ

P = {[−p_{0}, p_{0}], I_{k} = [−p_{k}, −p_{k−1}[ ∪]p_{k−1}, p_{k}]}_{k≥1} where p_{k} = 2^{k}  (2)

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

P = {[−p_{0}, p_{0}], I_{k} = [−p_{k}, −p_{k−1}[ ∪]p_{k−1}, p_{k}]} where log p_{k} ≫ log p_{k−1} (3)

grow with a speed that forces regularity. Put p_{k} = 2^{k!} for a simple explicit example. We have precisely

Proposition 3.2 Let E ⊆ ℤ, P = {I_{k}} a partition of ℤ and E_{k} = E ∩ I_{k}. Then log E_{k} ≽ log I_{k} in the two following cases:

(i) if E has regular polynomial growth and P is partition (2); (ii) if E has polynomial growth and P is partition (3).

Proof. (i) Choose K and c > 1 such that E[2^{k}] ≥ cE[2^{k−1}] for k ≥ K. Then E[2^{k}] ≽ c^{k}. Thus

E_{k} = E[2^{k}] − E[2^{k−1}] ≥ (1 − c^{−1})E[2^{k}] ≽ c^{k} = 2^{k log_{2} c}.

(ii) In this case p_{k}^{ε} ≫ p_{k−1} for any ε > 0. Now there is ε > 0 such that

E_{k} = E[p_{k}] − E[p_{k−1}] ≽ p_{k}^{ε} ≽ I_{k} ^{ε}.

4 Sets that are Λ(p) for all p

In this section, we establish an improvement (Th. 4.7) of Katznelson’s statement [51, §2]. We first recall several known definitions and results. Λ(p) sets have a practical description in terms of unconditionality. We shall also use a combinatorial property that is more elementary than [86, 1.6(b)]: to this end, write Z_{s}^{m} for the following set of arithmetic relations.

Z_{s}^{m} = {ζ ∈ ℤ^{∗}^{m} : ζ_{1} + … + ζ_{m} = 0 and |ζ_{1}| + … + |ζ_{m}| ≤ 2s}.

Note that Z_{s}^{1} and Z_{s}^{m} (m > 2s) are empty, and that every ζ ∈ Z_{s}^{2} is of form ζ_{1} · (1, −1): this is the identity relation.

Definition 4.1 Let 1 ≤ p < ∞, s ≥ 1 integer and E ⊆ ℤ.

(i) [49] E is an unconditional basic sequence in L^{p}(𝕋) if

sup_{±}∥∑_{n∈E} ±a_{n} e_{n}∥_{p} ≤ C∥∑_{n∈E} a_{n} e_{n}∥_{p}.

for some C. If C = 1 works, E is a 1-unconditional basic sequence in L^{p}(𝕋). (ii) [Chapter II, §2.2] E is s-independent if ∑ ζ_{i}q_{i} ≠ 0 for all 3 ≤ m ≤ 2s, ζ ∈ Z_{s}^{m} and distinct q_{1}, …, q_{m} ∈ E.

Proposition 4.2 Let 1 ≤ p < ∞, s ≥ 1 integer and E ⊆ ℤ.

(i) [86, proof of Th. 3.1] E is a Λ(max(p, 2)) set if and only if E is an unconditional basic sequence in L^{p}(𝕋). (ii) [Chapter II, Prop. 2.2.1] E is a 1-unconditional basic sequence in L^{2s}(𝕋) if and only if E is s-independent.

We need to introduce a second classical notion of unconditionality that rests on the Littlewood–Paley theory.

Definition 4.3 ([37]) Let P = {I_{k}} be a partition of ℤ in finite intervals. P is a Littlewood–Paley partition if for each 1 < p < ∞ there is a constant C_{p} such that

∀f ∈ L^{p}(𝕋) sup_{±}∥∑ ±f_{k}∥_{p} ≤ C_{p}∥f∥_{p}  with f̂_{k} = \{ \begin{matrix}f̂ & on \\ 0 off \end{matrix} I_{k}.  (4)

By Khinchin’s inequality, this means exactly that

∀f ∈ L^{p}(𝕋) ∥f∥_{p} ≈ ∥(∑ |f_{k}|^{2})^{1/2}∥_{p}.

In particular, the dyadic partition (2) and the gross partition (3) are Littlewood– Paley [61]. By Proposition 4.2 and (4), we obtain

Proposition 4.4 Let {I_{k}} be a Littlewood–Paley partition and E_{k} ⊆ I_{k}. If E_{k} is s-independent for each k, then E = ⋃ E_{k} is an unconditional basic sequence in L^{2s}(𝕋) and thus a Λ(2s) set.

We generalize now Katznelson’s Proposition 2.7.

Lemma 4.5 Let s ≥ 2 integer, E ⊆ ℤ finite and 0 ≤ ℓ ≤ E . Put δ_{n} = l/ E in Construction 2.4, so that all selectors ξ_{n} have same distribution. Then there is a constant C(s) that depends only on s such that

ℙ [E′ is s-dependent ] ≤ C(s)\frac{ℓ^{2s}}{E} .

Proof. We wish to compute the probability that there are 3 ≤ m ≤ 2s, ζ ∈ Z_{s}^{m} and distinct q_{1}, …, q_{m} ∈ E′ with ∑ ζ_{i}q_{i} = 0. As the number C(s) of arithmetic relations ζ ∈ Z_{s}^{m} (3 ≤ m ≤ 2s) is finite and depends on s only, it suffices to compute, for fixed m and ζ ∈ Z_{s}^{m}

ℙ [∃q_{1}, …, q_{m} ∈ E′ distinct : ∑ ζ_{i}q_{i} = 0]
= ℙ [∃q_{1}, …, q_{m−1} ∈ E′ distinct : −ζ_{m}^{−1} ∑_{i=1}^{m−1} ζ_{i}q_{i} ∈ E′ \ {q_{1}, …, q_{m−1}}]

= ℙ [ ⋃_{q_{1},...,q_{m−1} ∈E′ distinct} [−ζ_{m}^{−1} ∑_{i=1}^{m−1} ζ_{i}q_{i} ∈ E′ \ {q_{1}, …, q_{m−1}}]]

= ℙ [ ⋃_{q_{1},...,q_{m−1} ; ∈E distinct} [q_{m} = −ζ_{m}^{−1} ∑_{i=1}^{m−1} ζ_{i}q_{i} ∈ E \ {q_{i}}_{i=1}^{m−1} & ξ_{q_{1}} = … = ξ_{q_{m}} = 1]]

The union in the line above runs over

\frac{E !}{( E − m + 1)!} ≤ E ^{m−1}

(m − 1)-tuples. Further, the event in the inner brackets implies that m out of E selectors ξ_{n} have value 1: its probability is bounded by (ℓ/ E )^{m}. Thus

ℙ [E′ is s-dependent] ≤ C(s) max_{3≤m≤2s} E ^{m−1} \frac{ℓ^{m}}{E ^{m}} ≤ C(s)\frac{ℓ^{2s}}{E} .

The random method we shall use is the following random construction.

Construction 4.6 Let E ⊆ ℤ. Let {I_{k}} be a Littlewood–Paley partition and E_{k} = E ∩ I_{k}. Let (ℓ_{k})_{k≥1} with 0 ≤ ℓ_{k} ≤ E_{k} and put

ℙ [ξ_{n} = 1] = δ_{n} = ℓ_{k}/ E_{k} (n ∈ E_{k})

in Construction 2.4. Put E′_{k} = E′ ∩ I_{k}.

Theorem 4.7 Let E ⊆ ℤ have polynomial (vs. regular) growth and {I_{k}} be the gross (3) (vs. dyadic (2)) Littlewood–Paley partition. Do Construction 4.6. The following assertions are equivalent.

(i) log ℓ_{k} ≪ log I_{k} , i. e. log ℓ_{k} ≪ log p_{k} (vs. log ℓ_{k} ≪ k); (ii) E′ is almost surely a Λ(p) set for all p.

Proof. Note that by Proposition 3.2, there is a positive α such that E_{k} > I_{k} ^{α} for large k. (i) ⇒ (ii) Let s ≥ 2 be an arbitrary integer. By Proposition 4.5,

∑_{k=1}^{∞} ℙ [E′_{k} is s-dependent] ≤ C(s) ∑_{k=1}^{∞} \frac{ℓ_{k}^{2s}}{E_{k}} .

For each η > 0, ℓ_{k} ≤ I_{k} ^{η} for large k. Choose η < α/2s. Then ℓ_{k}^{2s}/ E_{k} ≤ I_{k} ^{2sη−α} for large k, and the series above converges since I_{k} ≽ 2^{k}. By the Borel–Cantelli lemma, E′_{k} is almost surely s-independent for large k. By Proposition 4.4, E′ is almost surely the union of a finite set and a Λ(2s) set. By [86, Th. 4.4(a)], E′ itself is almost surely a Λ(2s) set. (ii) ⇒ (i) If E′ is a Λ(2s) set, then by [86, Th. 3.5] or simply by [9, (1.12)], there is a constant C_{s} such that E′_{k} < C_{s} I_{k} ^{1/s}. As E′_{k} ∼ ℓ_{k} almost surely by the Law of Large Numbers (cf. the following Lemma 5.1), one has log ℓ_{k} ≪ log I_{k} .

Remark 4.8 As one can easily construct sets that grow as slowly as one wishes and nevertheless contain arbitrarily large intervals (see also [86, Th. 3.8] for an optimal statement), one cannot remove the adverb “almost surely” in Theorem 4.7(ii).

Remark 4.9 The right formulation of Katznelson’s Proposition 2.7 thus turns out to be the following. Let E = ℕ and I_{k} = ]p_{k−1}, p_{k}] with p_{k} > cp_{k−1} for some c > 1 in Construction 4.6. Then E′ is almost surely a Λ(p) set for all p if and only if log ℓ_{k} ≪ log p_{k}.

Remark 4.10 Theorem 4.7 shows that there are sets that are Λ(p) for all p of any given superpolynomial order of growth. This is optimal since sets with distribution E[t] ≽ t^{ε} fail the Λ(p) property for p > 2/ε by [86, Th. 3.5]. Such sets may also be constructed inductively by combinatorial means: see Section II.11 and [35, §II, (3.52)].

5 Equidistributed sets

In this section, we shall finally state and prove our principal result. To this end, we shall first generalize Bourgain’s Theorem 2.5 in order to get Theorem 5.4. The following lemma is Bernstein’s distribution inequality [4] and dates back to 1924.

Lemma 5.1 Let X_{1}, …, X_{n} be complex independent random variables such that

|X_{i}| ≤ 1 and 𝔼 X_{i} = 0 and 𝔼 |X_{1}|^{2} + … + 𝔼 |X_{n}|^{2} ≤ σ.  (5)

Then, for all positive a,

ℙ [|X_{1} + … + X_{n}| ≥ a] < 4 exp(−a^{2}/4(σ + a)).  (6)

Proof. Consider first the case of real random variables. By [1, (8b)],

ℙ [X_{1} + … + X_{n} ≥ a] < exp(a − (σ + a) log(1 + a/σ));

as log(1 − u) ≤ −u − u^{2}/2 for 0 ≤ u < 1,

ℙ [X_{1} + … + X_{n} ≥ a] < exp(−a^{2}/2(σ + a)).

One gets (6) since for complex z

|z| ≥ a ⟹ max(ℜz, −ℜz, ℑz, −ℑz) ≥ a/\sqrt{2}.

The next lemma corresponds to [8, Lemma 8.8] and is the crucial step in the estimation of the successive means of {e^{int}}_{n∈E′}. Note that its hypothesis is not on the individual δ_{n}, but on their successive sums σ_{k}: this is needed in order to cope with the irregularity of E.

Lemma 5.2 Let E = {n_{k}} ⊆ ℤ be ordered by increasing absolute value. Do Construction 2.4 and put σ_{k} = δ_{n_{1}} + … + δ_{n_{k}}. If σ_{k} ≫ log |n_{k}|, then almost surely

ψ(k) = ∥ \frac{1}{E′ ∩ \{n_{1}, …, n_{k}\}} ∑ e_{n}_{E′∩\{n_{1},...,n_{k}\}} −\frac{1}{σ_{k}} ∑_{j=1}^{k} δ_{n_{j}} e_{n_{j}} ∥_{∞} ⟶_{k→∞} 0. (7)

Proof. Note that

∑ e_{n}_{E′∩{n_{1},...,n_{k}}} = ∑_{j=1}^{k} ξ_{n_{j}} e_{n_{j}} , E′ ∩ {n_{1}, …, n_{k}} = ∑_{j=1}^{k} ξ_{n_{j}}.

Center the ξ_{n} by putting f = ∑_{j=1}^{k}(ξ_{n_{j}} − δ_{n_{j}}) e_{n_{j}}. Then

ψ(k) ≤ ∥( E′ ∩ {n_{1}, …, n_{k}}^{−1} − σ_{k}^{−1}) ∑_{j=1}^{k} ξ_{n_{j}} e_{n_{j}} ∥_{∞} + ∥σ_{k}^{−1}f∥_{∞}
≤ σ_{k}^{−1} |\frac{δ_{n_{1}} + … + δ_{n_{k}}}{ξ_{n_{1}} + … + ξ_{n_{k}}} − 1| ∑_{j=1}^{k} ξ_{n_{j}} + σ_{k}^{−1}∥f∥_{∞} ≤ 2σ_{k}^{−1}∥f∥_{∞}.

Let R = {t ∈ 𝕋 : t^{4|n_{k}|} = 1} and u ∈ 𝕋 such that |f(u)| = ∥f∥_{∞}. Let t ∈ R be at minimal distance of u: then |t − u| ≤ π/4|n_{k}|. By Bernstein’s theorem,

∥f∥_{∞} − |f(t)| = |f(u) − f(t)| ≤ |t − u| ∥f′∥_{∞} ≤ \frac{4}{5}∥f∥_{∞}

∥f∥_{∞} ≤ 5 sup_{t∈R} |f(t)|.

(For an optimal bound, cf. [68, §I.4, Lemma 8].) For each t ∈ R, the random variables X_{j} = (ξ_{n_{j}} − δ_{n_{j}}) e_{n_{j}}(t) satisfy (5), so that

ℙ [|f(t)| ≥ a] < 4 exp(−a^{2}/4(σ_{k} + a)).

As R = 4|n_{k}|,

ℙ [∥f∥_{∞} ≥ 5a] ≤ ℙ [sup_{t∈R} |f(t)| ≥ a] < 4|n_{k}| · 4 exp(−a^{2}/4(σ_{k} + a)).

Put a_{k} = (12σ_{k} log |n_{k}|)^{1/2}. Then a_{k} ≪ σ_{k}: therefore

ℙ [∥f∥_{∞} ≥ 5a_{k}] ≼ |n_{k}|^{−2}

and by the Borel–Cantelli lemma,

σ_{k}^{−1}∥f∥_{∞} ≼ a_{k}/σ_{k} ⟶ 0 almost surely.

Remark 5.3 The hypothesis in Lemma 5.2 contains implicitly a restriction on the lacunarity of E. If σ_{k} ≫ log |n_{k}|, then necessarily log |n_{k}| ≪ k and E[t] ≫ log t. In particular, E cannot be a Sidon set by [86, Cor. of Th. 3.6].

We now state and prove the equidistributed counterpart of Theorem 4.7.

Theorem 5.4 Let E = {n_{k}} ⊆ ℤ be equidistributed (vs. weakly), and ordered by increasing absolute value. Do Construction 2.4 and suppose that δ_{n_{j}} decreases with j. Put σ_{k} = δ_{n_{1}} +…+δ_{n_{k}}. If σ_{k} ≫ log |n_{k}|, then E′ is almost surely equidistributed (vs. weakly). Note that this is in particular the case if

(a) δ_{n_{k}} ≫ (|n_{k}| − |n_{k−1}|)/|n_{k−1}|;

(b) E has polynomial growth and δ_{n_{k}} ≫ k^{−1}.

Proof. Lemma 5.2 shows that almost surely (7) holds. It remains to show that

lim \frac{1}{σ_{k}} ∑_{j=1}^{k} δ_{n_{j}} e_{n_{j}} = lim \frac{1}{k} ∑_{j=1}^{k} e_{n_{j}},

i. e. that the matrix summing method (a_{k,j}) given by

a_{k,j} = \{ \begin{matrix}δ_{n_{j}}/σ_{k} & if j ≤ k \\ 0 & otherwise \end{matrix}

is regular and stronger than the Cesàro method C_{1} by arithmetic means. This is the case because a_{k,j} ≥ 0, ∑_{j} a_{k,j} = 1 and (cf. [100, §52, Th. I])

∀k ∑_{j} j|a_{k,j} − a_{k,j+1}| = ∑_{j} j(a_{k,j} − a_{k,j+1}) = 1 < ∞

since (a_{k,j}) decreases with j for each k.

(a) In this case δ_{n_{k}} ≫ log |n_{k}| − log |n_{k−1}| and thus σ_{k} ≫ log |n_{k}|.

(b) In this case, σ_{k} ≫ log k ≽ log |n_{k}|.

In conclusion, we obtain, by combining Theorems 4.7 and 5.4, the principal result of this chapter.

Corollary 5.5 Let E ⊆ ℤ be equidistributed (vs. weakly) and do Construction 4.6. Then E′ is almost surely Λ(p) for all p and at the same time equidistributed (vs. weakly) in the two following cases:

(i) E is a set of regular polynomial growth, {I_{j}} is the dyadic Littlewood–Paley partition (2) and one may choose {ℓ_{j}} such that 1 ≪ log ℓ_{j} ≪ j and ℓ_{j}/ E_{j} decreases eventually. (ii) E is a set of polynomial growth, {I_{j}} is the gross Littlewood–Paley partition (3) and one may choose {ℓ_{j}} such that ℓ_{j}/ E_{j} decreases eventually and ℓ_{j} ≫ log p_{j+1} while log ℓ_{j} ≪ log p_{j}. This is the case if we put p_{j} = 2^{j!} and ℓ_{j} = min((j +2)!, E_{j} ).

Proof. In each case log ℓ_{j} ≪ log I_{j} . Let us show that the hypothesis of Theorem 5.4 is verified. If n_{k} ∈ E_{j} ⊆ I_{j}, then |n_{k}| ≤ p_{j} and

σ_{k} ≥ ∑_{i=1}^{j−1} ∑_{n∈E_{i}} δ_{n} = ℓ_{1} + … + ℓ_{j−1}

and in each case ℓ_{j−1} ≫ log p_{j} − log p_{j−1}. Let us make sure in (ii) that our choice for p_{j} and ℓ_{j} is accurate. Indeed, there is an ε > 0 such that E_{j} ≽ 2^{εj!}. Thus (j + 2)! ≪ E_{j} and ℓ_{j} = (j + 2)! for large j. Note further that (j + 2)! ≫ (j + 1)! while log(j + 2)! ≪ j!. Finally

\frac{ℓ_{j+1}}{E_{j+1}} ≼ \frac{(j + 3)!}{2^{ε(j+1)!}} ≼ \frac{jℓ_{j}}{2^{ε(j+1)!}} ≪ \frac{ℓ_{j}}{2^{j!}} ≼ \frac{ℓ_{j}}{E_{j}} ,

so that ℓ_{j}/ E_{j} decreases eventually.

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Index of notation

B cardinal of B
X_{E} space of X-functions with spectrum in E
f̂ Fourier transform of f: f̂(n) = ∫ f(t) e_{−n}(t)dm(t)
\binom{x}{α} multinomial number, §2.3
⟨ζ, E⟩ pairing of the arithmetical relation ζ against the spectrum E, §3.1
u_{n} ≼ v_{n} |u_{n}| is bounded by C|v_{n}| for some C
u_{n} ≪ v_{n} u_{n} is negligible with respect to v_{n}
E[t] distribution function of E ⊆ ℤ: E[t] = E ∩ [−t, t]
(x_{rc}) the operator x on ℓ_{2} viewed as matrix
q_{1} → q_{2} segment from q_{1} to q_{2}, where q_{1}, q_{2} ∈ ℕ × ℕ
q_{1} ↓ q_{2} segment from q_{1} to q_{2}, where q_{1}, q_{2} ∈ ℕ × ℕ
1-(ubs) 1-unconditional basic sequence of characters, Def. 2.1.1(i)
A(𝕋) disc algebra 𝒞_{ℕ}(𝕋)
A_{n}, A_{n}^{m} sets of multi-indices viewed as arithmetic relations, §2.2
a.s. approximating sequence, Def. 4.1.1
B_{X} unit ball of the Banach space X
𝒞(𝕋) space of continuous functions on 𝕋
c_{0}(𝕋) space {f: 𝕋 → 𝕋 : ∀ε > 0 ∃A ⊆ 𝕋 finite |f| < ε outside A}
c.b. completely bounded
𝔻 set of real signs {−1, 1}
ΔT_{k} difference sequence of the T_{k}: ΔT_{k} = T_{k} − T_{k−1} (T_{0} = 0)
𝔼 X expectation of the random variable X
e_{n} character of 𝕋: e_{n}(z) = z^{n} for z ∈ 𝕋, n ∈ ℤ
e_{rc} matrix entry seen as operator on ℓ_{2}
(fdd) finite dimensional decomposition, Def. 4.1.1
H^{1}(𝕋) Hardy space L_{ℕ}^{1}(𝕋)
(ℐ_{n}) arithmetical property of almost independence, Def. 2.4.1
Id identity
i.i.d. independent identically distributed, §12.2
(𝒥_{n}) arithmetical property of block independence, Def. 7.1.2
ℒ(X) space of bounded linear operators on the Banach space X
L^{p}(𝕋) Lebesgue space of p-integrable functions on 𝕋
ℓ_{p}-(ap) p-additive approximation property, Def. 5.1.1
ℓ_{p}-(map) metric p-additive approximation property, Def. 5.1.1
Λ(p) Rudin’s class of lacunary sets, Def. 2.1.6

ℳ_{p} functional property of Fourier block p-additivity, Lemma 6.2.5(ii)
ℳ(𝕋) space of Radon measures on 𝕋
m[A] measure of A ⊆ 𝕋
(m_{p}(τ)) functional property of τ-p-additivity, Def. 5.3.1(i)
(m_{p}(T_{k})) functional property of commuting block p-additivity, Def. 5.3.1(ii)
osc f oscillation of f
ℙ [A] probability of the event A
𝒫(𝕋) space of trigonometric polynomials on 𝕋
π_{j} projection of X_{E}, E = {n_{k}}, onto X_{{n_{1},...,n_{j}}}
π_{F} projection of X_{E} onto X_{F}
𝕊 real (𝕊 = 𝔻) or complex (𝕊 = 𝕋) choice of signs
S^{p} Schatten class of operators
S_{I}^{p} entry subspace of S^{p}
(𝕋, dm) unit circle in ℂ with its normalized Haar measure
τ_{f} topology of pointwise convergence of the Fourier coefficients, Lemma 6.2.2(i)
(𝒰) functional property of Fourier block unconditionality, Def. 6.2.1
(u(τ)) functional property of τ-unconditionality, Def. 4.2.1(i)
(u(T_{k})) functional property of commuting block unconditionality, Def. 4.2.1(ii)
(uap) unconditional approximation property, Def. 4.1.1
(ubs) unconditional basic sequence, Def. 2.1.1
(umap) metric unconditional approximation property, Def. 4.1.1
(umbs) metric unconditional basic sequence, Def. 2.1.1
Z^{m}, Z_{n}^{m} sets of multi-indices viewed as arithmetic relations, §2.2

index of notation 97

Index

1-unconditional approximation

property for spaces L_{E}^{p}(𝕋), p even, 59

1-unconditional basic sequence of

characters, 27 in 𝒞(𝕋) and L^{p}(𝕋), p ∉ 2ℕ, 30, 32 in spaces L^{p}(𝕋), p even, 30, 59 on the Cantor group, 66 on the infinite torus, 66

1-unconditional basic sequence of

entries, 75, 76 1-unconditional (fdd), 53

for L_{E}^{4}(𝕋), 70 for spaces L_{E}^{p}(𝕋), p even, 54

almost i.i.d. sequence, 67 almost independence, 34 approximating sequence, 38 approximation property, 38 arithmetic sequences, 81 arithmetical relation, 29, 32, 33

Bernstein, Serge, 86 Binet, J. P. M., 57 birelation, 29 Bishop, Errett A., 48 Blei, Ron C., 44 block independent set of integers, 52 boundedly complete approximating

sequence, 41 Bourgain, Jean, 25, 80, 82 break, 39, 52

Cantor group, 28, 31, 66 Carleson, Lennart, 48 Casazza, Peter G., 38 Cesàro summing method, 88 commuting block unconditionality, 39 completely bounded 1-unconditional

basic sequence of entries, 75 completely bounded Λ(p) set, 78 completely bounded Schur multiplier,

72 completely bounded unconditional

basic sequence of entries, 75 completely bounded unconditionality

constant of sum sets, 77

complex vs. real, 28, 31, 52, 66, 76 connected p-gonal relation, 72 connected rectangular lines, 72 cotype, 41

Daugavet property, 48, 70 distribution function, 82 dyadic partition, 83

entry, 71 entry subspace of S^{p}, 72 equidistributed set, 81, 88 equimeasurability, 25 Erdős, Paul, 65, 82 Euler’s conjecture, 37, 57 Euler, Leonhard, 57 exponential growth, 65

Fibonacci sequence, 57 finite-dimensional decomposition, 38 Forelli, Frank, 24 Fourier block unconditionality, 48, 52 Frénicle de Bessy, Bernard, 37

geometric sequences, 36, 56, 58, 68, 81 Godefroy, Gilles, 40, 45 gross partition, 83

Hadamard set, 64 Hankel Schur multiplier, 77 Harcharras, Asma, 77 Hilbert set, 44, 54 Hindman, Neil, 65 homogeneous Banach space, 26

independent set of integers, 29 infinite difference set, 65 isometries on L^{p}, 32 Kadets, Vladimir M., 48 Kalton, Nigel J., 24, 38, 40, 42, 45 Katznelson, Yitzhak, 82 Kazhdan, David A., 65

Λ(p) set, 29, 55, 79 constant, 29 for all p, 85, 88 sufficient conditions, 78

Li, Daniel, 45, 54, 65, 81 Littlewood–Paley partition, 47, 84 Lust-Piquard, Françoise, 48

Malliavin, Paul, 82 matrix independent set of entries, 76 matrix summing method, 88 maximal density, 64

of block independent sets, 65 of independent sets, 64

metric 1-additive approximation

property for spaces 𝒞_{E}(𝕋), 50, 59 for subspaces of L^{1}, 45

metric p-additive approximation

property, 42, 45 for homogeneous Banach spaces,

50 for subspaces of L^{p}, 46

metric unconditional approximation

property, 38, 40 for 𝒞_{E}(𝕋) and L_{E}^{p}(𝕋), p ∉ 2ℕ, 54 for homogeneous Banach spaces,

49, 59 for spaces L_{E}^{p}(𝕋), p even, 54 on the Cantor group, 66

metric unconditional basic sequence,

27, 34, 59 metric unconditional (fdd), 38, 54 Meyer, Yves, 54 Murai, Takafumi, 36, 67

oscillation, 26

p-additive approximation property, 42 for spaces 𝒞_{E}(𝕋), 44 for spaces L_{E}^{p}(𝕋), 44

p-gon, 72

that does not span a surface, 73,

74 that returns on its track, 73, 74

p-gonal relation, 72 Pełczyński, Aleksander, 38 Peller, Vladimir V., 77 Pisier, Gilles, 66 Plotkin, A. I., 24 polynomial growth, 65, 82 polynomial sequences, 37, 57, 80, 81,

83 biquadrates, 37, 57 cubes, 37, 57 squares, 37, 57

prime numbers, 80, 81, 83 pseudo-independent set, 67

Rademacher functions, 66 Ramanujan, Srinivasa, 57 random construction, 82, 85 real vs. complex, 28, 31, 52, 66, 76 rectangular line, 72 regular polynomial growth, 82 relative multipliers, 28

interpolation, 28 renormings, 25 Riesz product, 60, 64, 67 Riesz set, 47 Rosenblatt, Murray, 24 Rosenthal set, 47, 79 Rosenthal, Haskell Paul, 23, 25 Rudin, Walter, 29, 48, 78

Schur multiplier, 72 Schur property, 1-strong, 45 semi-Riesz set, 48 shrinking approximating sequence, 41 Sidon set, 29, 35, 44, 66

constant, 29, 63, 64

with constant asymptotically 1,

34, 59, 67 similar rectangular lines, 73 smoothness, 25 Stein, Elias, 48 strong mixing, 24 subexponential growth, 82 sup-norm-partitioned sets, 44, 48 superexponential growth, 65 superpolynomial growth, 65, 86 symmetric sets, 56, 58

tail, 39, 52 τ-unconditionality, 39 transcendental numbers, 36, 57, 69, 70 unconditional approximation property,

38 for spaces 𝒞_{E}(𝕋), 47

for spaces L_{E}^{1}(𝕋), 47

unconditional basic sequence of

characters, 27 unconditional basic sequence of

entries, 75 unconditional (fdd), 38 unconditional skipped blocking

decompositions, 40 unconditionality constant, 27

in L^{4}(𝕋), 31

Vinogradov, 81

weakly equidistributed set, 81, 88 Werner, Dirk, 42, 45, 48 Weyl, Hermann, 80 Wojtaszczyk, Przemysław, 38