\documentclass[a4paper]{article} \usepackage{lmodern} \usepackage[T1]{fontenc} \usepackage[frenchb,german,british]{babel} \usepackage[colorlinks=true,breaklinks=true,citecolor=red,linkcolor=blue,urlcolor=red,unicode]{hyperref} \usepackage[hyperpageref]{backref} \renewcommand*{\backref}[1]{} \renewcommand*{\backrefalt}[4]{\ifcase #1\or(p.~#2).\else(pp.~#2).\fi} \newlength{\marge} \setlength{\marge}{26mm} \addtolength{\voffset}{\marge} \addtolength{\voffset}{-125pt} \addtolength{\hoffset}{\marge} \addtolength{\hoffset}{-125pt} \setlength{\textheight}{11in} \addtolength{\textheight}{-\marge} \addtolength{\textheight}{-\footskip} \addtolength{\textheight}{-10mm} \setlength{\textwidth}\paperwidth \addtolength{\textwidth}{-2\marge} \usepackage{makeidx,amsmath} \makeindex \usepackage{amssymb} \usepackage{mathrsfs} \def\stefan{Stefan Neuwirth\par Laboratoire de Math\'ematiques\par Universit\'e de Franche-Comt\'e\par 16, route de Gray, 25000 Besan\c{c}on\par neuwirth@math.univ-fcomte.fr\par} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{ldc}[thm]{Computational lemma} \newtheorem{prp}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{dfn}[thm]{Definition} \newtheorem{pbl}[thm]{Problem} \newtheorem{construction}[thm]{Construction} \newtheorem{thmsub}{Theorem}[subsection] \newtheorem{lemsub}[thmsub]{Lemma} \newtheorem{prpsub}[thmsub]{Proposition} \newtheorem{corsub}[thmsub]{Corollary} \newtheorem{dfnsub}[thmsub]{Definition} \newtheorem{ldcsub}[thmsub]{Computational lemma} \newtheorem{pblsub}[thmsub]{Problem} \newtheorem{constructionsub}[thmsub]{Construction} \newtheorem{thmf}{Th\'eor\`eme}[section] \newtheorem{lemf}[thmf]{Lemme} \newtheorem{ldcf}[thmf]{Lemme de calcul} \newtheorem{prpf}[thmf]{Proposition} \newtheorem{corf}[thmf]{Corollaire} \newtheorem{dfnf}[thmf]{Définition} \newtheorem{thmsubf}{Th\'eor\`eme}[subsection] \newtheorem{lemsubf}[thmsubf]{Lemme} \newtheorem{prpsubf}[thmsubf]{Proposition} \newtheorem{corsubf}[thmsubf]{Corollaire} \newtheorem{dfnsubf}[thmsubf]{D\'efinition} \newtheorem{pblsubf}[thmsubf]{Probl\`emem} \newtheorem{constructionsubf}[thmsubf]{Construction} \def\dem{{\it Proof.\/ }} \def\rem{\refstepcounter{thm} {\bf Remark \thethm\hskip1ex}} \def\remsub{ \refstepcounter{thmsub}{\bf Remark \thethmsub}\hskip1ex} \def\exa{ \addtocounter{thm}{1}{\bf Example \thethm}\hskip1ex} \def\exasub{ \addtocounter{thmsub}{1}{\bf Example \thethmsub}\hskip1ex} \def\quest{ \refstepcounter{thm}{\bf Question \thethm}\hskip1ex} \def\questsub{ \refstepcounter{thmsub}{\bf Question \thethmsub}\hskip1ex} \def\questf{ \refstepcounter{thmf}{\bf Question \thethmf}\hskip1ex} \def\questsubf{ \refstepcounter{thmsubf}{\bf Question \thethmsubf}\hskip1ex} \def\vskipa{\vskip\abovedisplayskip} \def\vskipb{\vskip\belowdisplayskip} \def\ie{{\it i.\,e.}\ } \def\vs{{\it vs.}\ } \def\af{{\it a fortiori}\/\ } \def\toll#1{\vtop{\ialign {##\crcr \rightarrowfill \crcr \noalign{\kern -1pt \nointerlineskip \vskip2pt } $\hfil \scriptstyle{\ #1\ } \hfil $\crcr }}} \def\({{\fam0\rm (}} \def\){\/{\fam0\rm )}} \def\]{\mathopen]} \def\[{\mathclose[} \mathchardef\hook="312C \def\longhook {\mathop{\hook\joinrel\relbar\joinrel\longrightarrow}\limits} \def\tol{\mathop{-\mkern-4mu\smash-\mkern-4mu\smash\longrightarrow}\limits} \def\downto{\downarrow} \def\imp{\Rightarrow} \def\Imp{\Longrightarrow} \def\ssi{\Leftrightarrow} \def\Ssi{\Longleftrightarrow} \def\ecks{\rule{2mm}{2mm}} \def\bull{\leavevmode\kern .1ex\vrule height 1ex width .9ex depth -.1ex \kern .8ex} \def\bloc{\bull} \def\eck{\nolinebreak\hspace{\fill}\ecks} \def\barre{\rule[-2.5pt]{0.8pt}{10pt}} \def\lmes{\kern0.7pt\barre\kern0.7pt} \def\rmes{\kern0.7pt\/\barre\kern0.7pt} \def\mes#1{\mbox{$\lmes #1 \rmes$}} \def\smes#1{|#1|} \def\bi#1#2{\bigg(\kern-0.5pt{{#1}\atop{#2}}\kern-0.5pt\bigg)} \def\bip#1#2{\left(\kern-3pt {{#1}\atop{#2}}\kern-2pt \right)} \def\se{\subseteq}\def\es{\supseteq} \def\HM#1{\setbox0=\hbox{#1}\dimen0=\wd0 #1 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\def\ds{\displaystyle} \def\hoch#1{\mathop{#1}\limits} \def\lst#1#2{#1_1,\dots,#1_{#2}} \def\lstf#1#2{#1_{#2},#1_{#2+1},\dots} \def\lstp#1#2#3{#1_{#2},\dots,#1_{#3}} \def\sm#1#2{#1_1+\dots+#1_{#2}} \def\bip#1#2{\left(\kern-3pt {{#1}\atop{#2}}\kern-2pt \right)} \def\smp#1#2#3{#1_{#2}+\dots+#1_{#3}} \def\bm#1{$(${\it\bf #1}$)$} \def\lap#1{$\ell_{#1}$-$(\!${\it ap}$)$}\def\lpap/{\lap{p}} \def\lmap#1{$\ell_{#1}$-$(\!${\it map}$)$}\def\lpmap/{\lmap{p}} \def\uap/{\mbox{$(\!${\it uap}$)$}}\def\ubp/{\mbox{$(\!${\it ubp}$)$}} \def\as/{\mbox{a.s.}}\def\sbd/{\mbox{$(\!${\it sbd\/}$)$}} \def\ubs/{\mbox{$(\!${\it ubs}$)$}}\def\umbs/{\mbox{$(\!${\it umbs}$)$}} \def\umap/{\mbox{$(\!${\it umap}$)$}} \def\ap/{\mbox{$(\!${\it ap}$)$}}\def\map/{\mbox{$(\!${\it map}$)$}} \def\fdd/{\mbox{$(${\it fdd\/}$)$}}\def\umfdd/{$(${\it umfdd\/}$)$} \def\SC#1{\mathscr{C}_{#1}(\T)} \def\SCE{\SC{E}} \def\SL#1#2{{\fam0 L}^{#1}_{#2}(\T)} \def\SLE#1{\SL{#1}{E}} \def\SLP#1{\SL{p}{#1}} \def\SLPE{\SLE{p}} \def\DX/{$X\in\{\SC{},\SLP{}:1\le p<\infty\}$} \def\PT#1{\mbox{$\mathscr{P}_{#1}(\T)$}} \def\PTE/{\PT{E}} \def\J#1{\mbox{$(\mathscr{J}_{#1})$}} \def\I#1{\mbox{$(\mathscr{I}_{#1})$}} \def\UP/{\mbox{$(\mathscr{U})$}} \def\X#1{{\langle\zeta,#1\rangle}}\def\XE{{\X E}} \def\EL#1{$\Lambda(#1)$} \def\ER/{\mbox{Rosenthal}} \def\DE/{\mbox{$E\subseteq\Z$}} \def\DEE/{\mbox{$E=\{n_k\}\subseteq\Z$}} \def\DEEE/{\mbox{$E=\{n_k\}_{k\ge1}\subseteq\Z$}} \def\DL/{\mbox{$\L/\subseteq\Z$}} \def\DLL/{\mbox{$\L/=\{\lambda_k\}\subseteq\Z$}} \def\DLLL/{\mbox{$\L/=\{\lambda_k\}_{k\ge1}\subseteq\Z$}} \def\DI/{$I\subseteq\N\times\N$} \def\ES#1{$\Sigma(#1)$} \def\ud/{equidistributed} \def\wud/{weakly equidistributed} \def\pstar{^{\scriptscriptstyle(\kern-1pt\lower0.5pt \hbox{$\scriptstyle *$}\kern-1pt)}} \def\ppstar{^{\scriptscriptstyle(\kern-1pt*\kern-1pt)}} \hyphenation{ ana-logon ap-pro-xi-ma-tion asymp-to-ti-cally Ba-nach bet-ween cha-rac-te-ri-ze cha-rac-te-ri-za-tion cha-r-ac-ters clas-si-que cor-res-pon-ding cor-res-pon-dante cru-ciaux Da-cun-ha de-com-po-si-tions den-si-ty de-via-tion en-lar-ging equi-dis-tri-bu-ted eve-ry dis-tin-gui-shes exac-te-ment fol-lo-wing Fou-rier ge-ne-ra-li-ze ge-ne-ra-li-zes Hada-mard in-con-di-tion-nel in-de-pen-d-en-ce in-de-pen-dent In-equa-li-ty In-for-ma-tique in-te-gra-ble iso-me-tric iso-me-tri-scher Lit-tle-wood Mathe-ma-tica ma-the-ma-ti-schen maxi-mum par-ti-cu-lar po-ly-no-mial pro-ba-bi-li-s-tic pro-ba-bi-liste pro-ba-bi-li-ty pro-per-ties pro-per-ty Ra-de-ma-cher re-a-li-zes re-s-tric-tion se-quen-ces sim-p-lest spa-ce sui-van-te theo-rem to-po-lo-gy Un-be-dingt-heit un-con-di-tio-nal un-con-di-tio-nal un-con-di-tio-na-li-ty uni-form-ly uni-que } \author{Stefan Neuwirth} \title{Metric unconditionality and Fourier analysis} \date{} \begin{document}\parindent=0pt \maketitle\vfil\selectlanguage{french} {\bf R\'esum\'e } Nous \'etudions plusieurs propri\'et\'es fonctionnelles d'inconditionnalit\'e iso\-m\'e\-trique et presqu'iso\-m\'etrique en les exprimant \`a l'aide de multiplicateurs. Parmi ceux-ci, la notion la plus g\'en\'erale est celle de ``propri\'et\'e d'approximation inconditionnelle m\'etrique''. Nous la caract\'erisons parmi les espaces de Banach de cotype fini par une propri\'et\'e simple d'``inconditionnalit\'e par blocs''. En nous ramenant \`a des multiplicateurs de Fourier, nous \'etudions cette propri\'et\'e dans les sous-espaces des espaces de Banach de fonctions sur le cercle qui sont engendr\'es par une suite de ca\-rac\-t\`eres $\e^{\ii nt}$. Nous \'etudions aussi les suites basiques inconditionnelles iso\-m\'etriques et presqu'isom\'etriques de caract\`eres, en particulier les ensembles de Sidon de constante asymptotiquement 1. Nous obtenons dans chaque cas des propri\'et\'es combinatoires sur la suite. La propri\'et\'e suivante des normes ${\fam0 L}^p$ est cruciale pour notre \'etude: si $p$ est un entier pair, $\int|f|^p=\int{|f^{p/2}|}^2=\sum|\widehat{f^{p/2}}(n)|{\vphantom{|f^{p/2}|}}^2$ est une expression polyno\-miale en les coefficients de Fourier de $f$ et $\bar f$. Nous proposons d'ailleurs une estimation pr\'ecise de la constante de Sidon des ensembles \`a la Hadamard. \vskip\baselineskip \selectlanguage{german}{\bf Zusammenfassung } Verschiedene funktionalanalytische isometrische und fast iso\-me\-tri\-sche Unbedingtheitseigenschaften werden mittels Multiplikatoren untersucht. Am allgemeinsten ist die metrische unbedingte Approximationseigenschaft gefasst. Wir charakterisieren diese f\"ur Banach\-r\"aume mit endlichem Kotyp durch eine einfache ``blockweise'' Unbedingtheit. Daraufhin betrachten wir genauer den Fall von Funktionenr\"aumen auf dem Einheitskreis, die durch eine Folge von Frequenzen $\e^{\ii nt}$ aufgespannt werden. Wir untersuchen isometrisch und fast isometrisch un\-be\-ding\-te Basisfolgen von Frequenzen, unter anderem Sidonmengen mit einer Konstante asymptotisch zu 1. F\"ur jeden Fall erhalten wir kombinatorische Eigenschaften der Folge. Die folgende Eigenschaft der ${\fam0 L}^p$ Normen ist entscheidend f\"ur diese Arbeit: Ist $p$ eine gerade Zahl, so ist $\int|f|^p=\int{|f^{p/2}|}^2=\sum|\widehat{f^{p/2}}(n)| {\vphantom{|f^{p/2}|}}^2$ ein polynomialer Ausdruck der Fourierkoeffizienten von $f$ und $\bar f$. Des weiteren erhalten wir eine genaue Absch\"atzung der Sidonkonstante von Hadamardfolgen. \vskip\baselineskip \selectlanguage{british}{\bf Abstract } We study several functional properties of isometric and almost isometric unconditionality and state them as a property of families of multipliers. The most general such notion is that of ``metric unconditional approximation property''. We characterise this ``\umap/'' by a simple property of ``block unconditionality'' for spaces with nontrivial cotype. We focus on subspaces of Banach spaces of functions on the circle spanned by a sequence of characters $\e^{\ii nt}$. There \umap/ may be stated in terms of Fourier multipliers. We express \umap/ as a simple combinatorial property of this sequence. We obtain a corresponding result for isometric and almost isometric basic sequences of characters. Our study uses the following crucial property of the ${\fam0 L}^p$ norm for even $p$: $\int|f|^p=\int{|f^{p/2}|}^2=\sum|\widehat{f^{p/2}}(n)|{\vphantom{|f^{p/2}|}}^2$ is a polynomial expression in the Fourier coefficients of $f$ and $\bar f$. As a byproduct, we get a sharp estimate of the Sidon constant of sets {\it \`a la}\/ Hadamard. \vfil\vfil\pagebreak \tableofcontents\vfill\pagebreak \section{A general introduction in French} \selectlanguage{french} \subsection{Position du probl\`eme} % Ce travail se situe au croisement de l'analyse fonctionnelle et de l'analyse harmonique. Nous allons donner des \'el\'ements de r\'eponse \`a la question g\'en\'erale suivante. \questf Quelle est la validit\'e de la repr\'esentation \begin{equation}\label{intro-these:q1} f\sim\sum \varrho_q\e^{\ii\vartheta_q}\e_q \end{equation} de la fonction $f$ comme s\'erie de fr\'equences $\e_q$ d'intensit\'e $\varrho_q$ et de phase $\vartheta_q$? Les r\'eponses seront donn\'es en termes de l'espace de fonctions $X\ni f$ et du spectre $E\supseteq\{q:\varrho_q>0\}$. Consid\'erons par exemple les deux questions classiques suivantes dans le cadre des espaces de Banach homog\`enes de fonctions sur le tore $\T$, des fr\'equences de Fourier $\e_q(t)=\e^{\ii qt}$ et des coefficients de Fourier $$\varrho_q\e^{\ii\vartheta_q}=\int\e_{-q}f=\widehat{f}(q).$$ \questsubf\label{intro:q1} Est-ce que pour les fonctions $f\in X$ \`a spectre dans $E$ $$\Bigl\| f-\sum_{|q|\le n}\varrho_q\e^{\ii\vartheta_q}\e_q \Bigr\|_X\tol_{n\to\infty}0\hbox{ ?}$$ Cela revient \`a demander: est-ce que la suite $\{\e_q\}_{q\in E}$ rang\'ee par valeur absolue $|q|$ croissante est une base de $X_E$? En d'autres termes, la suite des multiplicateurs idempotents relatifs $T_n:X_E\to X_E$ d\'efinie par $$T_n\e_q=\left\{\begin{array}{cl} \e_q&\hbox{si }|q|\le n\\ 0&\hbox{sinon} \end{array}\right.$$ est-elle uniform\'ement born\'ee sur $n$? Soit $E=\Z$. Un \'el\'ement de r\'eponse classique est le suivant. %Soit $E=\Z$. La norme des $T_n$ se calcule: $$ \|T_n\|_{{\fam0 L}^2(\sT)\to{\fam0 L}^2(\sT)}=1\ ,\ \|T_n\|_{{\fam0 L}^1(\sT)\to{\fam0 L}^1(\sT)}= \|T_n\|_{\mathscr{C}(\sT)\to\mathscr{C}(\sT)}\asymp\log n. $$ On sait de plus que les $T_n$ sont aussi uniform\'ement born\'es sur ${\fam0 L}^p(\T)$, $1
0$, un sous-ensemble $F\se E$ fini tel que $$\Bigl\| \sum_{q\in E\setminus F}\varrho_q\e^{\ii\vartheta_q}\e_q \Bigr\|_X\le(1+\eps)\Bigl\| \sum_{q\in E\setminus F}\varrho_q\e_q \Bigr\|_X\hbox{ ?} $$ Dans le cas $X=\mathscr{C}(\T)$, cela signifiera que $E$ est un ensemble de constante de Sidon ``asymptotiquement 1''. De m\^eme, peut-on choisir pour chaque $\eps>0$ un ensemble fini $F$ tel que pour tout choix de signe ``r\'eel'' $\pm$ $$ \Bigl\| \sum_{q\in E\setminus F}\pm a_q\e_q \Bigr\|_X\le(1+\eps)\Bigl\| \sum_{q\in E\setminus F}a_q\e_q \Bigr\|_X\hbox{ ?} $$ Toutes ces questions s'agr\`egent autour d'un fait bien connu: sommer la s\'erie de Fourier de $f$ est une tr\`es mauvaise mani\`ere d'approcher la fonction $f$ d\`es que l'erreur consid\'er\'ee n'est pas quadratique. On sait qu'il est alors utile de rechercher des m\'ethodes de sommation plus lisses, c'est-\`a-dire d'autres suites approximantes plus r\'eguli\`eres. Il s'agit l\`a de suites d'op\'erateurs de rang fini sur $X_E$ qui approchent ponctuellement l'identit\'e de $X_E$. Nous pourrons toujours supposer que ces op\'erateurs sont des multiplicateurs. Une premi\`ere question est la suivante. \questsubf\label{intro:q7} \ Existe-t-il une suite approximante $\{T_n\}$ de multiplicateurs idempotents? Cela revient \`a demander: existe-t-il une d\'ecomposition de $X_E$ en sous-espaces $X_{E_k}$ de dimension finie avec \begin{equation}\label{intro:fdd} X_E=\bigoplus X_{E_k}\quad\hbox{et}\quad A_k:X_E\to X_{E_k}\ ,\ \e_q\mapsto \left\{\begin{array}{cl} \e_q&\hbox{si }q\in E_k\\ 0&\hbox{sinon} \end{array}\right. \end{equation} telle que la suite des $T_n=A_1+\dots+A_n$ est uniform\'ement born\'ee sur $n$? Soit $E=\Z$. Alors la r\'eponse est identique \`a la r\'eponse de la question \ref{intro:q1}. Mais nous pouvons produire dans ce cadre plus g\'en\'eral des d\'ecompositions inconditionnelles de $X_E$ en r\'eponse \`a la question suivante. \questsubf Pour quels espaces $X$ et spectres $E$ existe-t-il une d\'ecom\-po\-si\-tion comme ci-dessus telle que la famille des multiplicateurs \begin{equation}\label{intro:fddi} \sum_{k=1}^n\epsilon_kA_k\quad\hbox{avec } n\ge1\hbox{ et }\epsilon_k=\pm 1 \end{equation} est uniform\'ement born\'ee? Littlewood et Paley ont montr\'e que la partition de $\Z$ en $\Z=\bigcup E_k$ avec $E_0=\{0\}$ et $E_k=\{j:2^{k-1}\le |j|<2^k\}$ donne une d\'ecomposition inconditionnelle des espaces ${\fam0 L}^p(\T)$ avec $1
0$ il existe une suite approximante $\{T_n\}$ sur $X_E$ telle que $$ \sup_{\hbox{\scriptsize signes }\epsilon_n} \Bigl\|\sum\epsilon_n(T_n-T_{n-1})\Bigr\|_X\le1+\eps $$ En termes fonctionnels, $X_E$ a-t-il la propri\'et\'e d'approximation inconditionnelle m\'etrique? Il faudra distinguer le cas des signes complexes et r\'eels. \subsection{Propri\'et\'e d'approximation inconditionnelle m\'e\-tri\-que} Comme nos questions distinguent les choix de signe r\'eel et complexe, nous proposons pour la fluidit\'e de l'expos\'e de fixer un choix de signes $\U$ qui sera $\U=\T=\{\epsilon\in\C:|\epsilon|=1\}$ dans le cas complexe et $\U=\D=\{-1,1\}$ dans le cas r\'eel. Seule la question \ref{intro:q10} n'impose pas au pr\'ealable de forme particuli\`ere \`a la suite de multiplicateurs qui est cens\'ee r\'ealiser la propri\'et\'e consid\'er\'ee. Afin d'\'etablir un lien entre la \umap/ et la structure du spectre $E$, nous faisons le d\'etour par une \'etude g\'en\'erale de cette propri\'et\'e dans le cadre des espaces de Banach s\'eparables. \subsubsection{Amorce et queue d'un espace de Banach} Peter G.\ Casazza et Nigel J.\ Kalton ont d\'ecouvert le crit\`ere suivant: \begin{prpsubf} Soit $X$ un espace de Banach s\'eparable. $X$ a la \umap/ si et seulement s'il existe une suite approximante $\{T_k\}$ telle que $$ \sup_{\epsilon\in\sU}\|T_k+\epsilon(\Id-T_k)\|_{\mathscr{L}(X)} \toll{k\to\infty}1. $$ \end{prpsubf} Ceci exprime que la constante d'inconditionnalit\'e entre l'amorce $T_kX$ et la queue $(\Id-T_k)X$ de l'espace $X$ s'am\'eliore asymptotiquement jusqu'\`a l'optimum pour $k\to\infty$. La \umap/ s'exprime de mani\`ere plus \'el\'ementaire encore si l'on choisit d'autres notions adapt\'ees d'amorce et de queue. Nous proposons en particulier la d\'e\-fi\-ni\-tion suivante. \begin{dfnsubf} Soit $\tau$ une topologie d'espace vectoriel topologique sur $X$. $X$ a la propri\'et\'e $(u(\tau))$ de $\tau$-inconditionnalit\'e si pour chaque $x\in X$ et toute suite born\'ee $\{y_j\}$ $\tau$-nulle l'oscillation $$ \osc_{\epsilon\in\sU}\|x+\epsilon y_j\|_X= \sup_{\delta,\epsilon\in\sU} \bigl(\|x+\epsilon y_j\|-\|x+\delta y_j\|\bigr) $$ forme elle-m\^eme une suite nulle. \end{dfnsubf} Nous avons alors le th\'eor\`eme suivant. \begin{thmsubf}\label{intro:THM} Soit $X$ un espace de Banach s\'eparable de cotype fini avec la propri\'et\'e $(u(\tau))$. Si $X$ admet une suite approximante $\{T_k\}$ inconditionnelle et commutative telle que $T_kx\mathop{\to}\limits^\tau x$ uniform\'ement sur la boule unit\'e $B_X$, alors des combinaisons convexes successives $\{U_j\}$ de $\{T_k\}$ r\'ealisent la \umap/. \end{thmsubf} {\it Esquisse de preuve.\/ } On construit ces combinaisons convexes successives par le biais de d\'ecompositions skipped blocking. En effet, la propri\'et\'e $(u(\tau))$ a l'effet suivant sur $\{T_k\}$. Pour chaque $\eps>0$, il existe une sous-suite $\{S_k=T_{n_k}\}$ telle que toute suite de blocs $S_{b_k}-S_{a_k}$ obtenue en sautant les blocs $S_{a_{k+1}}-S_{b_k}$ se somme de mani\`ere $(1+\eps)$-inconditionnelle.\\ Soit $n\ge1$. Pour chaque $j$, $1\le j\le n$, la suite de blocs obtenue en sautant $S_{kn+j}-S_{kn+j-1}$ pour $k\ge0$ est $(1+\eps)$-inconditionnelle. Il s'agit alors d'estimer la moyenne sur $j$ de ces suites de blocs. On obtient une suite approximante et l'hypoth\`ese de cotype fini permet de contr\^oler l'apport des blocs saut\'es.\\ Alors $X$ a la \umap/ parce que $n$ et $\eps$ sont arbitraires.\eck \subsubsection{Amorce et queue en termes de spectre de Fourier} Lorsqu'on consid\`ere l'espace invariant par translation $X_E$, une amorce et une queue naturelle sont les espaces $X_F$ et $X_{E\setminus G}$ pour $F$ et $G$ des sous-ensembles finis de $E$. Nous avons concr\`etement le lemme suivant. \begin{lemsubf} $X_E$ a $(u(\tau_f))$, o\`u $\tau_f$ est la topologie $$ f_n\mathop{\to}\limits^{\tau_f}0\ \Ssi\ \forall k\ \widehat{f_n}(k)\to0 $$ de convergence simple des coefficients de Fourier, si et seulement si $E$ est bloc-inconditionnel dans $X$ au sens suivant: quels que soient $\eps>0$ et $F\se E$ fini, il existe $G\se E$ fini tel que pour $f\in B_{X_F}$ et $g\in B_{X_{E\setminus G}}$ $$\osc_{\epsilon\in\sU}\|f+\epsilon g\|_X= \sup_{\delta,\epsilon\in\sU} \bigl(\|f+\epsilon g\|-\|f+\delta g\|\bigr)\le\eps.$$ \end{lemsubf} Le th\'eor\`eme \ref{intro:THM} s'\'enonce donc ainsi dans ce contexte particulier. \begin{thmsubf}\label{intro:bloc} Soit $E\se\Z$ et $X$ un espace de Banach homog\`ene de fonctions sur le tore $\T$. Si $X_E$ a la \umap/, alors $E$ est bloc-inconditionnel dans $X$. Inversement, si $E$ est bloc-inconditionnel dans $X$ et de plus $X_E$ a la propri\'et\'e d'approximation inconditionnelle et un cotype fini, alors $X_E$ a la \umap/. En particulier, on a \begin{itemize} \item[$(i)$]Soit $1
0$ et $F\se E\cap[-n,n]$. Soit $l$ tel que $|n_l|\ge \pi n/\eps$
et $G=\{\lst n{l-1}\}$. Soit $f\in B_{\mathscr{C}_F}$ et $g\in
B_{\mathscr{C}_{E\setminus G}}$. Alors $g(t+\pi/n_l)=-g(t)$ par
hypoth\`ese et
$$ |f(t+\pi/n_l)-f(t)|\le \pi/|n_l|\cdot\|f'\|_\infty\le\pi
n/|n_l|\le\eps $$
par l'in\'egalit\'e de Bernstein. Alors, pour un certain $u\in\T$
\begin{eqnarray*}
\|f-g\|_\infty&=&|f(u)+g(u+\pi/n_l)|\\
&\le&|f(u+\pi/n_l)+g(u+\pi/n_l)|+\eps\\
&\le&\|f+g\|_\infty+\eps.
\end{eqnarray*}
Donc $E$ est bloc-inconditionnel au sens r\'eel.\eck
En particulier, soit la suite g\'eom\'etrique $G=\{3^k\}$. Alors
$\mathscr{C}_G(\T)$ et $\mathscr{C}_{G\cup-G}(\T)$ ont la \umap/
r\'eelle.
\questsubf\label{intro:q2.1}
Qu'en est-il de la \umap/ complexe et qu'en est-il de la suite
g\'eom\'etrique $G=\{2^k\}$?
\subsection{Norme de multiplicateurs et conditions
combinatoires}\label{intro:nmcc}
Nous proposons ici une m\'ethode uniforme pour r\'epondre aux
questions \ref{intro:q4}, \ref{intro:q5}, \ref{intro:q6},
\ref{intro:q9} et \ref{intro:q10}. En effet, les questions
\ref{intro:q4}, \ref{intro:q5} et \ref{intro:q6} reviennent \`a
\'evaluer l'oscillation de la norme
$$
\Theta(\epsilon,a)=
\|\epsilon_0a_0\e_{r_0}+\dots+\epsilon_ma_m\e_{r_m}\|_X.
$$
La question \ref{intro:q9} revient \`a \'evaluer l'oscillation de la
norme
\begin{eqnarray*}
\Psi(\epsilon,a)&=&\Theta((\overbrace{1,\dots,1}^j,
\overbrace{\epsilon,\dots,\epsilon}^{m-j}),a)\\
&=&\|a_0\e_{r_0}+\dots+a_j\e_{r_j}+\epsilon a_{j+1}\e_{r_{j+1}}+\dots+
\epsilon a_m\e_{r_m}\|_X
\end{eqnarray*}
Par le th\'eor\`eme \ref{intro:bloc}, la question \ref{intro:q10}
revient \`a \'etudier cette m\^eme expression dans le cas particulier
o\`u on fait un saut de grandeur arbitraire entre $r_j$ et
$r_{j+1}$.\\
Dans le cas des espaces $X={\fam0 L}^p(\T)$, $p$ entier pair, ces normes sont
des polyn\^omes en $\epsilon$, $\epsilon^{-1}$, $a$ et $\bar a$. Dans
le cas des espaces $X={\fam0 L}^p(\T)$, $p$ non entier pair, elles s'expriment
comme des s\'eries. Il n'y a pas moyen d'exprimer ces normes comme
fonction $\mathscr{C}^\infty$ pour $X=\mathscr{C}(\T)$.
Soit $X={\fam0 L}^p(\T)$. D\'eveloppons $\Theta(\epsilon,a)$. Posons
$q_i=r_i-r_0$. On peut supposer $\epsilon_0=1$ et $a_0=1$. Nous
utilisons la notation suivante:
$$
{x\choose\alpha}=
\frac{x(x-1)\cdots(x-n+1)}{\alpha_1!\alpha_2!\dots}\quad\hbox{pour
}\alpha\in\N^m\hbox{ tel que }\sum\alpha_i=n
$$
Alors, si $|a_1|,\dots,|a_m|<1/m$ lorsque $p$ n'est pas un entier pair
et sans restriction sinon,
\begin{eqnarray}
\nonumber
\Theta(\epsilon,a)
&=&\int\biggl|\sum_{n\ge0}\bip{p/2}n
\biggl(\sum_{i=1}^m\epsilon_ia_i\e_{q_i}
\biggr)^n\biggr|^2\\
\nonumber&=&\int\biggl|
\sum_{n\ge0}\bip{p/2}n
\sum_{\scriptstyle\alpha:\lst\alpha m\ge0
\atop\scriptstyle\sm\alpha m=n}
\bi n\alpha
\epsilon^\alpha a^\alpha
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}
\biggr|^2\\
\nonumber&=&\int\biggl|\sum_{\alpha\in\sN^m}
\bip{p/2}\alpha
\epsilon^\alpha a^\alpha
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}\biggr|^2\\
\nonumber&=&\sum_{R\in\mathscr{R}}
\biggl|\sum_{\alpha\in R}
\bip{p/2}\alpha\epsilon^\alpha a^\alpha
\biggr|^2\\
&=&
\sum_{\alpha\in\sN^m}
{\bip{p/2}\alpha}^2|a|^{2\alpha}+
\sum_{\scriptstyle\alpha\ne\beta\in\sN^m\atop\scriptstyle\alpha\sim\beta}
\bip{p/2}\alpha\bip{p/2}\beta
\epsilon^{\alpha-\beta}a^\alpha \bar a^\beta\label{intro:gc:res}
\end{eqnarray}
o\`u $\mathscr{R}$ est la partition de $\N^m$ induite par la relation
d'\'equivalence
$$\alpha\sim\beta\ssi\sum\alpha_iq_i=\sum\beta_iq_i.$$
Nous pouvons r\'epondre imm\'ediatement aux questions \ref{intro:q4}
et \ref{intro:q5} pour $X={\fam0 L}^p(\T)$.
\subsubsection[Suites basiques 1-inconditionnelles complexes]
{Question \ref{intro:q4}: suites basiques 1-inconditionnelles complexes}
\label{sec:intro:q4}
Soient $r_0,\dots r_m$ sont choisis dans $E$, alors \eqref{intro:gc:res}
doit \^etre constante pour $a\in\{|z|<1/m\}^m$ et $\epsilon\in\T^m$. Cela
veut dire que pour tous $\alpha\ne\beta\in\N^m$,
$$
\sum\alpha_iq_i\ne\sum\beta_iq_i\quad\hbox{ou}\quad
\bip{p/2}\alpha\bip{p/2}\beta=0.
$$
\bloc
Si $p$ n'est pas un entier pair, alors
$\bip{p/2}\alpha\bip{p/2}\beta\ne0$ pour tous $\alpha,\beta\in\N^m$ et
on a les relations arithm\'etiques suivantes sur $q_1,q_2,0$:
\begin{eqnarray*}
\overbrace{q_1+\dots+q_1}^{|q_2|}=
\overbrace{q_2+\dots+q_2}^{|q_1|}&&\mbox{si }q_1q_2>0;\\
\overbrace{q_1+\dots+q_1}^{|q_2|}+\overbrace{q_2+\dots+q_2}^{|q_1|}
=0&&\mbox{sinon.}
\end{eqnarray*}
Il suffit donc de prendre respectivement
$$(\alpha,\beta)=\bigl((|q_2|,0,\dots),(|q_1|,0,\dots)\bigr)$$
et
$$(\alpha,\beta)=\bigl((|q_2|,|q_1|,0,\dots),(0,\dots)\bigr)$$
pour conclure que $\{r_0,r_1,r_2\}$ n'est pas une suite
basique 1-inconditionnelle complexe dans ${\fam0 L}^p(\T)$ si $p$
n'est pas un entier pair.
\bloc
Si $p$ est un entier pair, $\bip{p/2}\alpha\bip{p/2}\beta=0$ si et
seulement si
$$\sum\alpha_i>p/2\quad\hbox{ou}\quad\sum\beta_i>p/2.$$
On obtient que
$E$ est une suite basique 1-inconditionnelle dans ${\fam0 L}^p(\T)$ si et
seulement si $E$ est ``$p$-ind\'ependant'', c'est-\`a-dire que
$\sum\alpha_i(r_i-r_0)\ne\sum\beta_i(r_i-r_0)$ pour tous
$r_0,\dots,r_m\in E$ et $\alpha\ne\beta\in\N^m$ tels que
$\sum\alpha_i,\sum\beta_i\le p/2$. Cette condition est \'equivalente
\`a: tout entier $n\in\Z$ s'\'ecrit de mani\`ere au plus unique comme
somme de $p/2$ \'el\'ements de $E$.
\subsubsection[Suites basiques 1-inconditionnelles r\'eelles]
{Question \ref{intro:q5}: suites basiques 1-inconditionnelles r\'eelles}
Les suites basiques 1-inconditionnelles r\'eelles et complexes
co\"\i ncident et la r\'eponse \`a la question \ref{intro:q5} est
identique \`a la r\'eponse \`a la question \ref{intro:q4}. En effet,
d\`es qu'une relation arithm\'etique $\sum(\alpha_i-\beta_i)q_i$
p\`ese sur $E$, on peut supposer que $\alpha_i-\beta_i$ est impair
pour au moins un $i$ en simplifiant la relation par le plus grand
diviseur commun des $\alpha_i-\beta_i$. Mais alors \eqref{intro:gc:res}
n'est pas une fonction constante pour $\epsilon_i$ r\'eel.
Cette propri\'et\'e est propre au tore $\T$. En effet, par exemple la
suite des fonctions de Rademacher est 1-inconditionnelle r\'eelle
dans $\mathscr{C}(\D^\infty)$, alors que sa constante
d'inconditionnalit\'e complexe est $\pi/2$.
\subsubsection[Suites basiques inconditionnelles m\'etriques]
{Question \ref{intro:q6}: suites basiques inconditionnelles m\'etriques}
On peut m\^eme tirer des cons\'equences utiles du calcul de
\eqref{intro:gc:res} dans le cas pres\-qu'iso\-m\'e\-tri\-que. Il faut pour cela
prendre la pr\'ecaution suivante qui permet un passage \`a la
limite. Soit $0<\varrho<1/m$. Alors
$$\big\{\Theta\colon\U^m\times\{|z|\le\varrho\}^m\to\R^+:q_1,\dots,
q_m\in\Z^m\big\}$$
est un sous-ensemble relativement compact de
$\mathscr{C}^\infty(\U^m\times\{|z|\le\varrho\}^m)$. Il en d\'ecoule
que si $E$ est une suite basique inconditionnelle m\'etrique, alors
certains coefficients de \eqref{intro:gc:res} deviennent arbitrairement
petits lorsque $q_1,\dots,q_m$ sont choisis grands.
Donnons deux cons\'equences de ce raisonnement.
\begin{prpsubf}
Soit $E\se\Z$.
\begin{itemize}
\item[$(i)$]Soit $p$ un entier pair. Si $E$ est une suite basique inconditionnelle
m\'etrique r\'eelle, alors $E$ est en fait une suite basique
1-inconditionnelle complexe \`a un ensemble fini pr\`es.
\item[$(ii)$]Si $E$ est un ensemble de Sidon de constante asymptotiquement 1,
alors
$$\XE=\sup_{G\se E\hbox{\scriptsize\ fini}}\,
\inf\bigl\{|\zeta_1p_1+\dots+\zeta_mp_m|:
\lst p m\in E\setminus G\hbox{ distincts}\bigr\}>0$$
pour tout $m\ge1$ et $\zeta\in{\Z^*}^m$.
\end{itemize}
\end{prpsubf}
On peut exprimer cette derni\`ere propri\'et\'e en disant que la
relation arithm\'etique $\zeta$ ne persiste pas sur $E$.
\subsubsection[Propri\'et\'e d'approximation inconditionnelle m\'etrique]
{Question \ref{intro:q10}: propri\'et\'e d'approximation inconditionnelle m\'e\-tri\-que}
On peut appliquer la technique du paragraphe pr\'ec\'edent en
observant que si $X_E$ a la \umap/, alors
$$
\osc_{\epsilon\in\sU}\Psi(\epsilon,a)
\toll{\lstp r{j+1}m\in E\to\infty}0.
$$
\begin{dfnsubf}
$E$ a la propri\'et\'e $\J{n}$ de bloc-ind\'ependance si pour tout
$F\se E$ fini il existe $G\se E$ fini tel que si un $k\in\Z$ admet
deux repr\'esentations comme somme de $n$ \'el\'ements de
$F\cup(E\setminus G)$
$$\sm pn=k=\sm{p'}n,$$
alors
$$\mes{\{j:p_j\in F\}}\quad\hbox{et}\quad\mes{\{j:p'_j\in F\}}$$
sont \'egaux \(choix de signes complexe $\U=\T$\) ou de m\^eme parit\'e
\(choix de signes r\'eel $\U=\D$\).
\end{dfnsubf}
\begin{thmsubf}
Soit $E\se\Z$.
\begin{itemize}
\item [$(i)$]Si $X={\fam0 L}^p(\T)$, $p$ entier pair, alors ${\fam0 L}^p_E(\T)$ a la \umap/ si et
seulement si $E$ satisfait $\J{p/2}$.
\item [$(ii)$]Si $X={\fam0 L}^p(\T)$, $p$ non entier pair, ou $X=\mathscr{C}(\T)$, alors
$X_E$ a la \umap/ seulement si $E$ satisfait
$$\XE=\sup_{G\se E\hbox{\scriptsize\ fini}}\,
\inf\bigl\{|\zeta_1p_1+\dots+\zeta_mp_m|:
\lst p m\in E\setminus G\hbox{ distincts}\bigr\}>0$$
pour tout $m\ge1$ et $\zeta\in{\Z^*}^m$ tel que $\sum\zeta_i$ est non
nul \(cas complexe\) ou impair \(cas r\'eel\).
\end{itemize}
\end{thmsubf}
On obtient la hi\'erarchie suivante.
$$
\parbox{13mm}{$\SCE$ a\\\umap/}\imp
\parbox{25mm}{$\SLPE$ a \umap/,\\$p$ non entier pair}
\imp\dots\imp
\parbox{15mm}{$\SLE{2n+2}$\\ a \umap/}\imp
\parbox{14mm}{$\SLE{2n}$ a\\\umap/}
\imp\dots\imp\parbox{13mm}{$\SLE{2}$ a\\\umap/.}
$$
Nous pouvons r\'epondre \`a la question \ref{intro:q2.1}. Soit
$G=\{j^k\}$ avec $j\in\Z\setminus\{-1,0,1\}$ et consid\'erons
$\zeta=(j,-1)$. Alors $\langle\zeta,G\rangle=0$. Donc
$\mathscr{C}_G(\T)$ n'a pas la \umap/ complexe. $\mathscr{C}_G(\T)$ n'a
pas la \umap/ r\'eelle si $j$ est pair.
\subsubsection{Deux exemples}
\`A l'aide de nos conditions arithm\'etiques, nous sommes \`a m\^eme
de prouver la proposition suivante.
\begin{prpsubf}
Soit $\sigma>1$ et $E$ la suite des parties enti\`eres de
$\sigma^k$. Alors les assertions suivantes sont \'equivalentes.
\begin{itemize}
\item [$(i)$]$\sigma$ est un nombre transcendant.
\item [$(ii)$]${\fam0 L}^p_E(\T)$ a la \umap/ complexe pour tout $p$ entier pair.
\item [$(iii)$]$E$ est une suite basique inconditionnelle m\'etrique dans chaque
${\fam0 L}^p(\T)$, $p$ entier pair.
\item [$(iv)$]Pour chaque $m$ donn\'e, la constante de Sidon des sous-ensembles
\`a $m$ \'el\'ements de queues de $E$ est asymptotiquement 1.
\end{itemize}
\end{prpsubf}
Nous obtenons aussi la proposition suivante.
\begin{prpsubf}
Soit $E$ la suite des bicarr\'es. ${\fam0 L}^p_E(\T)$ a la \umap/ r\'eelle
seulement si $p=2$ ou $p=4$.
\end{prpsubf}
{\it Preuve.\/ }
$E$ ne satisfait pas la propri\'et\'e de bloc-ind\'ependance $\J{3}$
r\'eelle. En effet, Ramanujan a d\'ecouvert l'\'egalit\'e suivante
pour tout $n$:
$$
(4n^5-5n)^4+(6n^4-3)^4+(4n^4+1)^4=
(4n^5+n)^4+(2n^4-1)^4+3^4.\eqno{\ecks}
$$
\subsection{Impact de la croissance du spectre}
\label{sec:intro:croiss}
Nous d\'emontrons de mani\`ere directe le r\'esultat positif suivant.
\begin{thmsubf}
Soit $E=\{n_k\}\se\Z$ tel que $n_{k+1}/n_k\to\infty$. Alors la suite
des projections associ\'ee \`a $E$ r\'ealise la \umap/ complexe dans
$\SCE$ et $E$ est un ensemble de Sidon de constante asymptotiquement
1. Dans l'hypoth\`ese o\`u les rapports $n_{k+1}/n_k$ sont tous
entiers, la r\'eciproque vaut.
\end{thmsubf}
\begin{corsubf}
Alors $E$ est une suite basique inconditionnelle m\'etrique dans tout
espace de Banach homog\`ene $X$ de fonctions sur $\T$. De plus, $X_E$ a la
\umap/ complexe.
\end{corsubf}
{\it Esquisse de preuve.\/ }
Nous prouvons concr\`etement que si $n_{k+1}/n_k\to\infty$, alors quel
que soit $\eps>0$ il existe $l\ge1$ tel que pour toute fonction
$f=\sum a_k\e_{n_k}$
\begin{equation}\label{p6}
\|f\|_\infty\ge(1-\eps)\Bigl(\Bigl\|\sum_{k\le
l}a_k\e_{n_k}\Bigr\|_\infty+\sum_{k>l}|a_k|\Bigr).
\end{equation}
Cela revient \`a dire que la suite $\{\pi_k\}$ de projections
associ\'ee \`a la base $E$ r\'ealise la $1/(1-\eps)$-\uap/. Pour
obtenir l'in\'egalit\'e \eqref{p6}, on utilise une r\'ecurrence
bas\'ee sur l'id\'ee suivante.
Soit $u\in\T$ tel que $\|\pi_kf\|_\infty=|\pi_kf(u)|$.
Il existe alors $v\in\T$ tel que
$$
|u-v|\le\pi/|n_{k+1}|\quad\mbox{et}\quad
|\pi_kf(u)+a_{k+1}\e_{n_{k+1}}(v)|=\|\pi_kf\|_\infty+|a_{k+1}|.$$
De plus, dans ce cas,
$$
|\pi_kf(u)-\pi_kf(v)|\le
|u-v|\,\|\pi_kf'\|_\infty\le
\pi|n_k/n_{k+1}|\,\|\pi_kf\|_\infty.
$$
En r\'esum\'e, $a_{k+1}\e_{n_{k+1}}$ a le m\^eme argument que $\pi_kf$
tr\`es pr\`es du maximum de $|\pi_kf|$, et $\pi_kf$ varie peu.
Mais alors
\begin{eqnarray*}
\|\pi_kf(t)+a_{k+1}\e_{n_{k+1}}\|_\infty&\ge&|\pi_kf(v)+a_{k+1}\e_{n_{k+1}}(v)|\\
&\ge&\|\pi_kf\|_\infty+|a_{k+1}|-\pi|n_k/n_{k+1}|\|\pi_kf\|_\infty\\
&=&(1-\pi|n_k/n_{k+1}|)\|\pi_kf\|_\infty+|a_{k+1}|.
\end{eqnarray*}
On obtient \eqref{p6} en r\'eit\'erant cet argument.\eck
Notre technique donne d'ailleurs l'estimation suivante de la constante
de Sidon des ensembles de Hadamard.
\begin{corsubf}
Soit $E=\{n_k\}\se\Z$ et $q>\sqrt{\pi^2/2+1}$. Si $|n_{k+1}|\ge
q|n_k|$, alors la constante de Sidon de $E$ est inf\'erieure ou
\'egale \`a $1+\pi^2/(2q^2-2-\pi^2)$.
\end{corsubf}
Nous prouvons que cette estimation est optimale au sens o\`u
l'ensemble $E=\{0,1,q\}$, $q\ge2$, a pour constante
d'inconditionnalit\'e r\'eelle dans $\SC{}$
$$\bigl(\cos(\pi/(2q)\bigr)^{-1}\ge1+\pi^2/8\,q^{-2}.$$
%\baselineskip=15.6pt minus 1.2pt
\selectlanguage{british}
\section{Introduction}
%
%
%
%
%
We study isometric and almost isometric counterparts to the following
two properties of a separable Banach space $Y$:
\vskipa {\bf (ubs) } $Y$ is the closed span of an unconditional basic
sequence;
\vskipa{\bf (uap) } $Y$ admits an unconditional finite dimensional
expansion of the identity.\vskipb
We focus on the case of translation invariant spaces of functions on
the torus group $\T$, which will provide us with a bunch of natural
examples. Namely, let $E$ be a subset of $\Z$ and $X$ be one of the
spaces $\SLP{}$ $(1\le p<\infty)$ or $\SC{}$. If $\{\e^{{\rm
i}nt}\}_{n\in E}$ is an unconditional basic sequence (\ubs/ for
short) in $X$, then $E$ is known to satisfy strong conditions of
lacunarity: $E$ must be in Rudin's class \EL{q}, $q=p\vee2$, and a
Sidon set respectively. We raise the following question: what kind of
lacunarity is needed to get the following stronger property:
\vskipa{\bf (umbs) } $E$ is a metric unconditional basic sequence in
$X$: for any $\eps>0$, one may lower its unconditionality constant to
$1+\eps$ by removing a finite set from it.\vskipb
In the case of $\SC{}$, $E$ is a \umbs/ exactly when $E$ is a Sidon set
with constant asymptotically 1.
In the same way, call $\{T_k\}$ an approximating sequence (\as/ for
short) for $Y$ if the $T_k$'s are finite rank operators that tend
strongly to the identity on $Y$; if such a sequence exists, then $Y$
has the bounded approximation property. Denote by $\Delta
T_k=T_k-T_{k-1}$ the difference sequence of $T_k$. Following
Rosenthal\index{Rosenthal, Haskell Paul}
(see \cite[\S1]{fe80}), we then say that $Y$ has the unconditional
approximation property (\uap/ for short) if it admits an \as/
$\{T_k\}$ such that for some $C$
\begin{equation}\label{intro:uap}
\biggl\|\sum_{k=1}^n
\epsilon_k\Delta T_k
\biggr\|_{\mathscr{L}(Y)} \le C
\qquad\mbox{for all $n$ and scalar $\epsilon_k$ with
$|\epsilon_k|=1$.}
\end{equation}
By the uniform boundedness principle, \eqref{intro:uap} means
exactly that $\sum\Delta T_ky$ converges unconditionally for all $y\in
Y$. We now ask the following question: which conditions on $E$ do
yield the corresponding almost isometric (metric for short) property,
first introduced by Casazza and Kalton \cite[\S3]{ck91}?
\vskipa{\bf (umap) } The span $Y=X_E$ of $E$ in $X$ has the metric
unconditional approximation property: for any $\eps>0$, one may lower
the constant $C$ in \eqref{intro:uap} to $1+\eps$ by choosing an
adequate \as/ $\{T_k\}$.\vskipb
Several kinds of metric, \ie almost isometric properties have been
investigated in the last decade (see \cite{hww93}). There is a common
feature to these notions since
Kalton's\index{Kalton, Nigel J.}
\cite{ka93}: they can be reconstructed from a
corresponding interaction between some break and some tail of the
space. We prove that \umap/ is characterised by almost
1-unconditionality between a specific break and tail, that we coin
``block unconditionality''.\vskipb
Property \umap/ has been studied by Li \cite{li96} for $X=\SC{}$. He
obtains remarkably large examples of such sets $E$, in particular
Hilbert sets. Thus, the second property seems to be much weaker than
the first (although we do not know whether $\SCE$ has \umap/ for all \umbs/
$E$ in $\SC{}$: for sets of the latter kind, the natural sequence of
projections realises \uap/ in $\SCE$, but we do not know whether it achieves
\umap/).\vskipb
In fact, both problems lead to strong arithmetical conditions on $E$
that are somewhat complementary to the property of quasi-independence
(see \cite[\S3]{pi81}). In order to obtain them, we apply
Forelli's\index{Forelli, Frank} \cite[Prop.\ 2]{fo64} and
Plotkin's\index{Plotkin, A. I.} \cite[Th.\ 1.4]{pl74} techniques
in the study of isometric operators on ${\fam0 L}^p$: see Theorem
\ref{mub:thm} and Lemma \ref{umap:lem}. This may be done at once for
the projections associated to basic sequences of characters. In the
case of general metric unconditional approximating sequences, however,
we need a more thorough knowledge of their connection with the structure
of $E$: this is the duty of Theorem \ref{sbd:thm}. As in
Forelli's
and Plotkin's results, we obtain that the spaces
$X=\SLP{}$ with $p$
an even integer play a special r\^ole. For instance, they are the
only spaces which admit 1-unconditional basic sequences \DE/ with
more than two elements: see Proposition \ref{mub:isom}.\vskipb
There is another fruitful point of view: we may consider elements of
$E$ as random variables on the probability space $(\T,dm)$. They have
uniform distribution and if they were independent, then our questions
would have trivial answers. In fact, they are strongly dependent: for
any $k,l\in\Z$, Rosenblatt's\index{Rosenblatt, Murray} \cite{ro56}
strong mixing coefficient\index{strong mixing}
%
$$
%
\sup\bigl\{|m[A\cap B]-m[A]m[B]|: A\in\sigma(\e^{\ii kt})\mbox{ and
}B\in\sigma(\e^{\ii lt}) \bigr\}
%
$$
%
has its maximum value, $1/4$. But lacunarity of $E$ enhances their
independence in several weaker senses (see \cite{be90}). Properties
\umap/ and \umbs/ can be seen as an expression of almost independence
of elements of $E$ in the ``additive sense'', \ie when appearing in
sums. We show their relationship to the notions of pseudo-independence
(see \cite[\S4.2]{mu82}) and almost i.i.d.\ sequences (see
\cite{be87}).\vskipb
The gist of our results is the following: almost isometric properties
for spaces $X_E$ in ``little'' Fourier analysis may be read as a
smallness property of $E$. They rely in an essential way on the
arithmetical structure of $E$ and distinguish between real and complex
properties. In the case of $\SL{2n}{}$, $n$ integer, these arithmetical
conditions are in finite number and turn out to be sufficient, because
the norm of trigonometric polynomials is a polynomial expression in
these spaces. Furthermore, the number of conditions increases with
$n$ in that case. In the remaining cases of $\SLP{}$, $p$ not an even
integer, and $\SC{}$, these arithmetical conditions are infinitely many
and become much more coercive. In particular, if our properties are
satisfied in $\SC{}$, then they are satisfied in all spaces $\SLP{}$,
$1\le p<\infty$. \vskipb
We now turn to a detailed discussion of our results: in Section
\ref{sect:mub}, we first characterise the sets $E$ and values $p$
such that $E$ is a 1-unconditional basic sequence in $\SLP{}$ (Prop.\
\ref{mub:isom}). Then we show how to treat similarly the almost
isometric case and obtain a range of arithmetical conditions \I{n} on
$E$ (Th.\ \ref{mub:thm}). These conditions turn out to be identical
whether one considers real or complex unconditionality: this is
surprising and in sharp contrast to what happens when $\T$ is replaced
by the Cantor group. They also do not distinguish amongst $\SLP{}$
spaces with $p$ not an even integer and $\SC{}$, but single out $\SLP{}$
with $p$ an even integer: this property does not ``interpolate''. This
is similar to the phenomena of
equimeasurability\index{equimeasurability} (see
\cite[introduction]{ko91}) and $\mathscr{C}^\infty$-smoothness of
norms\index{smoothness}
(see \cite[Chapter V]{dgz93}). These facts may also be appreciated
from the point of view of natural renormings\index{renormings} of the
Hilbert space
$\SLE{2}$.\vskipb
In Section \ref{sect:ex}, of purely arithmetical nature, we give
many examples of 1-uncon\-ditional and metric unconditional basic
sequences through an investigation of property \I{n}. As expected
with lacunary series, number theoretic conditions show up (see
especially Prop.\ \ref{mub:trans}).\vskipb
In Section \ref{sect:block}, we first return to the general case of
a separable Banach space $Y$ and show how to connect the metric
unconditional approximation property with a simple property of ``block
unconditionality''. Then a skipped blocking technique invented by
Bourgain\index{Bourgain, Jean} and
Rosenthal\index{Rosenthal, Haskell Paul} \cite{br80}
gives a canonical way to construct an \as/ that realises
\umap/ (Th.\ \ref{block:thm}).\vskipb
In Section \ref{sect:lpmap}, we introduce the $p$-additive approximation
property \lpap/ and its metric counterpart, \lpmap/. It may be
described as simply as \umap/. Then we connect \lpmap/ with the work
of Godefroy, Kalton, Li and Werner \cite{kw95,gkl96} on
subspaces of ${\fam0 L}^p$ which are almost isometric to $\ell_p$.\vskipb
Section \ref{sect:tis} focusses on \uap/ and \umap/ in the case of
translation invariant subspaces $X_E$. The property of block
unconditionality may then be expressed in terms of ``break'' and
``tail'' of $E$: see Theorem \ref{sbd:thm}.\vskipb
In Section \ref{sect:umap}, we proceed as in Section
\ref{sect:mub} to obtain a range of arithmetical conditions \J{n}
for \umap/ and metric unconditional \fdd/ (Th.\ \ref{umap:thm} and
Prop.\ \ref{umap:prp:fdd}). These conditions are similar to \I{n},
but are decidedly weaker: see Proposition \ref{arith:csq}$(i)$. This
time, real and complex unconditionality differ; again spaces $\SLP{}$
with even $p$ are singled out.\vskipb
In Section \ref{sect:arith}, we continue the arithmetical
investigation begun in Section \ref{sect:ex} with property \J{n} and
obtain many examples for the 1-un\-con\-di\-tio\-nal and the metric
unconditional approximation property.\vskipb
However, the main result of Section \ref{sect:positif}, Theorem
\ref{positif:thm}, shows how a rapid (and optimal) growth condition on
$E$ allows avoiding number theory in any case considered. We therefore
get a new class of examples for \umbs/, in particular Sidon sets of
constant asymptotically 1, and \umap/. We also prove that
$\SC{\{3^k\}}$ has real \umap/ and that this is due to the oddness of
$3$ (Prop.\ \ref{res:geo}). A sharp estimate of the Sidon constant of
Hadamard sets is obtained as a byproduct (Cor.\ \ref{positif:cor}).
\vskipb
Section \ref{sect:comb} uses combinatorial tools to give some rough
information about the size of sets $E$ that satisfy our arithmetical
conditions. In particular, we answer a question of Li \cite{li96}: for
$X=\SC{}$ and for $X=\SLP{}$, $p\ne2,4$, the maximal density $d^*$ of
$E$ is zero if $X_E$ has \umap/ (Prop.\ \ref{comb:thm}). For
$X=\SL{4}{}$, our technique falls short of the expected result: we
just know that if $\SL{4}{E\cup\{a\}}$ has \umap/ for every $a\in\Z$,
then $d^*(E)=0$.\vskipb
Section \ref{sect:proba} is an attempt to describe the relationship
between these notions and probabilistic independence. Specifically the
Rademacher and Steinhaus sequences show the way to a connection
between metric unconditionality and the almost i.i.d.\ sequences of
\cite{be87}. We note further that the arithmetical property
\I{\infty} of Section \ref{sect:mub} is equivalent to Murai's
\cite[\S4.2]{mu82} property of pseudo-independence.\vskipb
In Section \ref{sect:resume}, we collect our results on metric
unconditional basic sequences of characters and \umap/ in translation
invariant spaces. We conclude with open questions. \vskipb
{\bf Notation and definitions } Sections \ref{sect:mub},
\ref{sect:tis}, \ref{sect:umap} and \ref{sect:positif} will take place
in the following framework. $(\T,dm)$ denotes the compact abelian
group $\{z\in\C:|z|=1\}$ endowed with its Haar measure $dm$; $m[A]$ is
the measure of a subset $A\se\T$. Let $\D=\{-1,1\}$. $\U$ will denote
either the complex ($\U=\T$) or real ($\U=\D$) choice of signs. For a
real function $f$ on $\U$, the oscillation\index{oscillation} of $f$ is
$$
\osc_{\epsilon\in\sU}f(\epsilon)= \sup_{\epsilon\in\sU}f(\epsilon)-
\inf_{\epsilon\in\sU}f(\epsilon).
$$
We shall study homogeneous\index{homogeneous Banach space} Banach
spaces $X$ of functions on $\T$
\cite[Chapter I.2]{ka68}, and especially the peculiar behaviour of the
following ones: $\SLP{}$ ($1\le p<\infty$), the space of $p$-integrable
functions with the norm $\|f\|_p=(\int|f|^pdm)^{1/p}$, and $\SC{}$, the
space of continuous functions with the norm
$\|f\|_\infty=\max\{|f(t)|:t\in\T\}$. $\mathscr{M}(\T)$ is the dual of
$\SC{}$ realised as Radon measures on \T.
The dual group $\{\e_n\colon z\mapsto z^n:n\in\Z\}$ of $\T$ is
identified with $\Z$. We write $\mes{B}$ for the cardinal of a set
$B$. For a not necessarily increasing sequence \DEEE/, let \PTE/ be
the space of trigonometric polynomials span\-ned by [the characters in]
$E$. Let $X_E$ be the translation invariant subspace of those
elements in $X$ whose Fourier transform vanishes off $E$: for all
$f\in X_E$ and $n\notin E$, $\widehat{f}(n)=\int
f(t)\e_{-n}(t)dm(t)=0$. $X_E$ is also the closure of \PTE/ in
homogeneous $X$ \cite[Th.\ 2.12]{ka68}. Denote by $\pi_k:X_E\to X_E$
the orthogonal projection onto $X_{\{\lst n k\}}$. It is given by
$$
\pi_k(f)= \widehat{f}(n_1)\e_{n_1}+\dots+\widehat{f}(n_k)\e_{n_k}.
$$
Then the $\pi_k$ commute. They form an \as/ for $X_E$ if and only
if $E$ is a basic sequence. For a finite or cofinite $F\se E$,
$\pi_F$ is similarly the orthogonal projection of $X_E$ onto $X_F$.
Sections \ref{sect:block} and \ref{sect:lpmap} consider the general
case of a separable Banach space $X$. $B_X$ is the unit ball of $X$
and $\Id$ denotes the identity operator on $X$. For a given sequence
$\{U_k\}$, its difference sequence is $\Delta U_k=U_k-U_{k-1}$ (where
$U_0=0$).
The functional notions of \ubs/, \umbs/
% and the unconditionality
%constants $C_p(E)$
are defined in \ref{mub:def}. The functional notions of \as/, \uap/
and \umap/ are defined in \ref{block:def}. Properties \lpap/ and
\lpmap/ are defined in \ref{str:def}. The functional property \UP/ of
block unconditionality is defined in \ref{block:block:def}. The sets
of arithmetical relations $\Zeta^m$ and $\Zeta_n^m$ are defined before
\ref{mub:isom}. The arithmetical properties \I{n} of almost
independence and \J{n} of block independence are defined in
\ref{mub:def:ar} and \ref{arith:def} respectively. The pairing $\XE$
is defined before \ref{mub:lim}.
\section{\texorpdfstring{Metric unconditional basic sequences of characters \umbs/}{Metric unconditional basic sequences of characters}}
\label{sect:mub}
\subsection{Definitions. Isomorphic case}
We start with the definition of metric
unconditional basic sequences (\umbs/ for short).
$\U=\T=\{\epsilon\in\C:|\epsilon|=1\}$ (\vs $\U=\D=\{-1,1\}$)
is the complex (\vs real) choice of signs.
%
\begin{dfnsub}\label{mub:def}
Let \DE/ and $X$ be a homogeneous Banach space on \T.
\begin{itemize}
\item [$(i)$]\cite{ka48} $E$ is an unconditional basic
sequence\index{unconditional basic sequence of characters} \ubs/
in $X$ if there is a constant $C$ such that
\begin{equation}\label{umbp:dfn}
\biggl\|\sum_{q\in G}\epsilon_qa_q\e_q\biggr\|_X\le
C\biggl\|\sum_{q\in G}a_q\e_q\biggr\|_X
\end{equation}
for all finite subsets $G\se E$, coefficients $a_q\in\C$ and signs
$\epsilon_q\in\T$ \(\vs $\epsilon_q\in\D$\). The infimum of such $C$
is the complex \(\vs real\) unconditionality\index{unconditionality constant}
%\index{complex unconditionality constant}
%\index{real unconditionality constant}
constant of $E$ in $X$. If $C=1$
works, then $E$ is a complex \(\vs real\)
1-\ubs/\index{1-unconditional basic sequence of characters} in
$X$.
\item [$(ii)$]$E$ is a complex \(\vs real\) metric unconditional basic
sequence\index{metric unconditional basic sequence} \umbs/ in $X$ if
for each $\eps>0$ there is a finite set $F$ such that
%$C^{\fam0 c}_p(E\setminus F)$
%\(\vs $C^{\fam0 r}_p(E\setminus F)$\)
the complex \(\vs real\) unconditionality constant of $E\setminus F$
is less than $1+\eps$.
\end{itemize}
\end{dfnsub}
%
Note that \Z\ itself is an \ubs/ in $\SLP{}$ if and only if $p=2$ by
Khinchin's inequality. The same holds in the
framework of the Cantor group\index{Cantor group} $\D^\infty$ and its dual group of Walsh
functions: their common feature with the $\e_n$ is that their modulus
is everywhere equal to 1 (see \cite{ke81}).
The following facts are folklore.
%
\begin{prpsub}
%
Let $Y$ be a Banach space.
\begin{itemize}
\item [$(i)$]If $\bigl\|\sum\epsilon_ky_k\bigr\|_Y\le C\bigl\|\sum
y_k\bigr\|_Y$ for all $\epsilon_k\in\T$ \(\vs $\epsilon_k\in\D$\),
then this holds automatically for all complex \(\vs real\)
$\epsilon_k$ with $|\epsilon_k|\le 1$.
\item [$(ii)$]Real\index{real vs.\ complex}\index{complex vs.\ real} and
complex unconditionality are isomorphically $\pi/2$-equivalent.
\end{itemize}
\end{prpsub}
%
\dem
$(i)$ follows by convexity. $(ii)$ Let us use the fact that the
complex unconditionality constant of the Rademacher sequence is $\pi/2$
\cite{se97}:
\begin{eqnarray*}
\sup_{\delta_k\in\sT}\Bigl\|\sum\delta_ky_k\Bigr\|_Y
&=&\sup_{y^*\in Y^*}\sup_{\delta_k\in\sT}\sup_{\epsilon_k=\pm1}
\Bigl|\sum\delta_k\langle y^*,y_k\rangle\epsilon_k\Bigr|\\&\le&
\pi/2\sup_{y^*\in Y^*}\sup_{\epsilon_k=\pm1}
\Bigl|\sum\langle y^*,y_k\rangle\epsilon_k\Bigr|=
\pi/2\sup_{\epsilon_k=\pm1}\Bigl\|\sum \epsilon_ky_k\Bigr\|_Y.
\end{eqnarray*}
Taking the Rademacher sequence in $\mathscr{C}(\D^\infty)$,
we see that $\pi/2$ is optimal.\eck\vskipb
%The constant $2$ above is indeed very bad in many cases.
%One can improve the constant $2$ above in certain cases: see
%\cite{se97}.
In fact, if \eqref{umbp:dfn} holds, then $E$ is a basis
of its span in $X$, which is $X_E$
% by Weierstra{\ss}'
%theorem
\cite[Th.\ 2.12]{ka68}.
We have the following relationship between
the unconditionality
constants of $E$ in $\SC{}$ and in a homogeneous Banach space $X$ on
$\T$. %$\SLPE$ ($1\le p<\infty$).:
%
\begin{prpsub}\label{mub:pinf}
%
Let \DE/ and $X$ be a homogeneous Banach space on $\T$.
\begin{itemize}
\item [$(i)$]The complex \(\vs real\) unconditionality constant of
$E$ in $X$ is at most the complex \(\vs real\)
unconditionality constant of $E$ in $\SC{}$.
\item [$(ii)$]If $E$ is a \ubs/ \(\vs 1-\ubs/, \umbs/\) in $\SC{}$,
then $E$ is a
\ubs/ \(\vs 1-\ubs/, \umbs/\) in $X$.
\end{itemize}
\end{prpsub}
%
This follows from the well-known (see {\it e.g.\/}
\cite{ha87})
%
\begin{lemsub}\label{sbd:multiplier}\index{relative multipliers}
%
Let \DE/ and $X$ be a homogeneous Banach space on $\T$. Let $T$ be a
multiplier on $\SCE$. Then $T$ is also a multiplier on $X_E$ and
$$\|T\|_{\mathscr{L}(X_E)}\le \|T\|_{\mathscr{L}(\mathscr{C}_E)}.$$
%$$
%\|T\|_{\mathscr{L}(\mathscr{C}_E)}=
%\supl_{1\le p<\infty}\|T\|_{\mathscr{L}({\fam0 L}^p_E)}.
%$$
%
\end{lemsub}
%
\dem
The linear functional $f\mapsto Tf(0)$ on $\SCE$ extends to a
measure $\mu\in\mathscr{M}(\T)$ such that
$\|\mu\|_\mathscr{M}=\|T\|_{\mathscr{L}(\mathscr{C}_E)}$. Let
$\check{\mu}(t)=\mu(-t)$. Then $Tf=\check{\mu}*f$ for $f\in\PTE/$ and
$$
\|T\|_{\mathscr{L}(X_E)}
\le\|\check{\mu}\|_\mathscr{M}
=\|T\|_{\mathscr{L}(\mathscr{C}_E)}.\eqno{\ecks}$$
%Furthermore, if $\|Tf\|_p\le C\|f\|_p$ for all $1\le p<\infty$,
%then $\|Tf\|_\infty\le C\|f\|_\infty$ by passing
%to the limit.
%\eck\vskipb
\questsub
There is no interpolation
theorem for such relative multipliers. The forthcoming
Theorem \ref{mub:thm} shows that there can be no
metric interpolation\index{relative multipliers!interpolation}.
Is it possible that one
cannot interpolate
multipliers at all between $\SLPE$ and $\SLE{q}$? \vskipb
Note that conversely, \cite{fo82} furnishes the example
of an \DE/ such that
the $\pi_k$ are uniformly
bounded on $\SLE{1}$ but not on $\SCE$.
It is known that $E$ is an \ubs/
in $\SC{}$ (\vs in $\SLP{}$) if and only if
it is a Sidon (\vs \EL{2\vee p}) set.
To see this, let us recall the relevant definitions.
%
\begin{dfnsub}\label{mub:sido}
%
Let \DE/.
\begin{itemize}
\item [$(i)$]\cite{ka57} $E$ is a Sidon\index{Sidon set} set if there is a
constant $C$ such that
$$
\sum_{q\in G}|a_q|\le C\biggl\|\sum_{q\in
G}a_q\e_q\biggr\|_\infty \mbox{ for all finite $G\se E$ and
$a_q\in\C$.}
$$
The infimum of such $C$ is $E$'s Sidon
constant\index{Sidon set!constant}.
\item [$(ii)$]\cite[Def.\ 1.5]{ru60} Let $p>1$. $E$ is a
\EL{p}\index{Lambda(p) set@\EL{p} set} set if there is a constant
$C$ such that $\|f\|_p\le C\|f\|_1$ for $f\in\PTE/$.
\end{itemize}
\end{dfnsub}
%
In fact, the Sidon constant of $E$ is the complex unconditionality
constant of $E$ in $\SC{}$. Thus $E$ is a complex \umbs/
in $\SC{}$ if and only if tails of $E$
have their Sidon
constant arbitrarily close to 1. We may also
say: $E$'s Sidon constant is
asymptotically 1.
Furthermore, $E$ is a \EL{2\vee p} set if and only if
$\SLPE=\SLE{2}$. Therefore \EL{2\vee p} sets are
\ubs/ in $\SLP{}$. Conversely, if $E$ is an
\ubs/ in $\SLP{}$, then by
Khinchin's
%\index{Khinchin's inequality}
inequality
$$
\biggl\|\sum_{q\in G} a_q\e_q\biggr\|_p^p\approx
\mbox{average}
\biggl\|\sum_{q\in G}\pm a_q\e_q\biggr\|_p^p\approx
\Bigl(\sum_{q\in G}|a_q|^2\Bigr)^{p/2}=
\biggl\|\sum_{q\in G} a_q\e_q\biggr\|_2^p
$$
for all finite $G\se E$ (see
\cite[proof of Th.\ 3.1]{ru60}). This shows also that the \EL{p}
set constant and the unconditionality constant in $\SLP{}$ are connected
{\it via}\/ the constants in Khinchin's inequality; whereas Sidon sets
have their unconditionality constant in $\SLP{}$ uniformly bounded, the
\EL{p} set constant\index{Lambda(p) set@\EL{p} set!constant}
of infinite sets grows at least like $\sqrt{p}$
\cite[Th.\ 3.4]{ru60}.
\subsection{Isometric case: 1-unconditional basic sequences of
characters}\label{ss:isom}
The corresponding isometric question: when is $E$ a complex
1-\ubs/? admits a rather easy answer. To
this end, introduce the following notation for arithmetical relations\index{arithmetical relation}:
let
$\Alpha_n=\bigl\{\alpha=\{\alpha_p\}_{p\ge1}:
\alpha_p\in\N\ \&\
\alpha_1+\alpha_2+\dots=n\bigr\}$. If
$\alpha\in \Alpha_n$, all but a finite number of the
$\alpha_p$ vanish and the multinomial number
$$\ds\bi n\alpha=\frac{n!}{\alpha_1!\alpha_2!\dots}$$
is well defined. Let
$\Alpha_n^m=\{\alpha\in \Alpha_n: \alpha_p=0\mbox{ for }p>m\}$.
Note that $\Alpha_n^m$ is finite.
We call $E$ $n$-independent\index{independent set of integers} if
every integer admits at most one representation as the sum of $n$
elements of $E$, up to a permutation. In terms of arithmetical
relations, this yields
$$
\sum\alpha_ip_i=\sum\beta_ip_i\imp\alpha=\beta\mbox{ for
$\alpha,\beta\in \Alpha_n^m$ and distinct $\lst pm\in E$}.
$$
This notion is studied in \cite{dr75} where it is called
birelation\index{birelation}. In Rudin's\index{Rudin, Walter}
\cite[\S$1.6(b)$]{ru60}
notation, the number $r_n(E;k)$ of
representations of $k\in\Z$ as a sum of $n$ elements of $E$ is at most
$n!$ for all $k$ if $E$ is $n$-independent (the converse if false).
This may also be expressed in the framework of arithmetical
relations
$$\Zeta^m=\{\zeta\in{\Z^*}^m:\sm\zeta m=0\}\quad\&\quad\Zeta_n^m=\{\zeta\in\Zeta^m:|\zeta_1|+\dots+|\zeta_m|\le 2n\}.$$
Note that $\Zeta_n^m$ is finite, and void if $m>2n$.
Then $E$ is $n$-independent if and only if
$$
\sum\zeta_ip_i\ne0\quad\hbox{for all }\zeta\in\Zeta_n^m
\hbox{ and distinct }\lst pm\in E.
$$
We shall prefer
to treat arithmetical relations in terms of $\Zeta_n^m$ rather
than $\Alpha_n^m$.
%
\begin{prpsub}\label{mub:isom}
\index{1-unconditional basic sequence of characters!in spaces $\SLP{}$, $p$ even}
\index{1-unconditional basic sequence of characters!in $\SC{}$ and $\SLP{}$, $p\notin2\N$}
Let \DE/.
\begin{itemize}
\item [$(i)$]$E$ is a complex 1-\ubs/ in $\SLP{}$, $p$ not an even
integer, or in $\SC{}$, if and only if $E$ has at most two elements.
\item [$(ii)$]If $p$ is an even integer, then $E$ is a complex 1-\ubs/
in $\SLP{}$ if and only if $E$ is $p/2$-independent. There is a
constant $C_p>1$ depending only on $p$, such that either $E$ is a
complex 1-\ubs/ in $\SLP{}$ or the complex unconditionality
constant of $E$ in $\SLP{}$ is at least $C_p$.
\end{itemize}
\end{prpsub}
%
\dem
%
$(i)$ By Proposition \ref{mub:pinf}$(ii)$, if $E$ is not a complex
1-\ubs/ in some $\SLP{}$, then neither in $\SC{}$.
Let $p$ be not an even integer. We may suppose
$0\in E$; let $\{0,k,l\}\se E$. If we had
$\|1+\mu a\e_k+\nu b\e_l\|_p=\|1+a\e_k+b\e_l\|_p$
for all $\mu,\nu\in\T$, then
\begin{eqnarray*}
\int|1+a\e_{k}+b\e_{l}|^pdm&=&
\int|1+\mu a\e_{k}+\nu b\e_{l}|^pdm(\mu)dm(\nu)dm\\
&=&
\int|1+\mu a+\nu b|^pdm(\mu)dm(\nu).
\end{eqnarray*}
Denoting by $\theta_i\colon(\epsilon_1,\epsilon_2)\mapsto\epsilon_i$
the projections of $\T^2$ onto $\T$, this would mean that
$\|1+a\e_{k}+b\e_{l}\|_p
=\|1+a\theta_1+b\theta_2\|
_{{\fam0 L}^p(\sT^2)}$
for all $a,b\in\C$. By
\cite[Th.\ I]{ru76},
$(\e_{k},\e_{l})$ and $(\theta_1,\theta_2)$
would have the
same distribution. This is false, since
$\theta_1$ and $\theta_2$ are independent random variables
while $\e_{k}$ and $\e_{l}$ are not.
$(ii)$
Let $\lst qm\in E$ be distinct and
$\lst\epsilon m\in\T$.
By the multinomial formula for the power $p/2$ and
Bessel--Parseval's formula, we get
\begin{eqnarray}
\nonumber
\lefteqn{\Biggl\|\sum_{i=1}^m\epsilon_ia_i\e_{q_i}\Biggr\|_p^p
=\int\Biggl|\sum_{\alpha\in \Alpha_{p/2}^m}
\bip{p/2}\alpha\prod_{i=1}^m(\epsilon_ia_i)^{\alpha_i}
\e_{\lower1pt\hbox{$\Sigma$}\alpha_iq_i}\Biggr|^2dm}\\
\nonumber&=&\sum_{A\in\mathscr{R}_q}\Biggl|
\sum_{\alpha\in A}
\bip{p/2}\alpha
\prod_{i=1}^m(\epsilon_ia_i)^{\alpha_i}\Biggr|^2\\
&=&\sum_{\alpha\in \Alpha_{p/2}^m}
{\bip{p/2}\alpha}^2\prod_{i=1}^m|a_i|^{2\alpha_i}
\label{cp>1}+
%\sum_{A\in\mathscr{R}_q}\sum_{\alpha\ne\beta\in A}
\sum_{\scriptstyle\alpha\ne\beta\in \Alpha_{p/2}^m\atop
\scriptstyle\alpha\sim\beta}
\bip{p/2}\alpha\bip{p/2}\beta
\prod_{i=1}^m\epsilon_i^{\alpha_i-\beta_i}
a_i^{\alpha_i}\overline{a_i}^{\beta_i},
\end{eqnarray}
where $\mathscr{R}_q$ is the partition of $\Alpha_{p/2}^m$
induced by the equivalence relation
$\alpha\sim\beta\Leftrightarrow
\sum\alpha_iq_i=\sum\beta_iq_i$. If $E$ is
$p/2$-independent, the second sum in \eqref{cp>1}
is void and $E$ is a
1-\ubs/.
Furthermore, suppose $E$ is not $p/2$-independent and
let $\lst qm\in E$ be a minimal set of distinct elements of $E$
such that there are $\alpha,\beta\in \Alpha_{p/2}^m$ with
$\alpha\sim\beta$. Then $m\le p$. Take $a_i=1$ in
the former computation: then the clearly nonzero
oscillation of
\eqref{cp>1} for $\lst\epsilon m\in\T$ does only
depend on $\mathscr{R}_q$ and thus is finitely valued.
This yields $C_p$.
%
\eck\vskipb
\exasub\index{unconditionality constant!in $\SL{4}{}$}
Let us treat explicitly the case $p=4$. If $E$ is not $2$-independent,
then one of the two following arithmetic relations occurs on $E$:
$$
2q_1=q_2+q_3\quad\hbox{or}\quad q_1+q_2=q_3+q_4.
$$
In the first case, we may assume $q_2 0$.
\item [$(ii)$]$X$ has \lpmap/.
\end{itemize}
\end{prpsub}
%
\dem
$(ii)\imp(i)$ is in Proposition \ref{appen:l1}.
For $(i)\imp(ii)$, it suffices
to prove that any subspace $Z$ of $\ell_p$
with the approximation property has \lpmap/.
As $Z$ is reflexive, $Z$ admits a commuting shrinking \as/
$\{R_n\}$. Let $i$ be the injection of $Z$ into $\ell_p$. Let
$\{P_n\}$ be the sequence of projections associated to the
natural basis of $\ell_p$. It is also an \as/ for $\ell_{p'}$. Thus
%
$$
\|i^*P_n^*x^*-R_n^*i^*x^*\|_{Z^*}\to0\qquad
\mbox{for any }x^*\in\ell_{p'}.
$$
As before,
there are convex combinations $\{C_n\}$ of
$\{P_n\}$ and $\{D_n\}$ of $\{R_n\}$ such that
$\|C_ni-iD_n\|\to0$.
The convex combinations are finite and may be chosen not to
overlap, so that for each $n\ge1$ there is
$m>n$ such that
$$
\|C_nx+(\Id -C_m)x\|=\bigl(\|C_nx\|^p+\|(\Id -C_m)x\|^p\bigr)^{1/p}
$$
for $x\in\ell_p$.
Thus $Z$ satisfies the property $(m_p(D_n))$.
Following the lines of \cite[Lemma 1]{fe80}, we observe that
$\{D_n\}$ is a commuting unconditional \as/ since $\{P_n\}$ is.
By Theorem \ref{block:strong:thm}, $Z$
has \lpmap/.\eck
\section{\texorpdfstring{\uap/ and \umap/ in translation invariant spaces}{(uap)
and (umap) in translation invariant spaces}}\label{sect:tis}
Recall that $\U$ is a subgroup of $\T$.
If $\U=\D=\{-1,1\}$, the following applies to real \umap/.
If $\U=\T=\{\epsilon\in\C:|\epsilon|=1\}$, it applies to complex
\umap/.
\subsection{General properties. Isomorphic case}
$\SLP{}$ spaces $(1 0$ and
finite $F\se E$, there is a finite $G\se E$ such that for $f\in
B_{X_F}$ and $g\in B_{X_{E\setminus G}}$
%
\begin{equation}\label{block:bloc}
%
\osc_{\epsilon\in\sU}\|\epsilon f+g\|_X\le\eps.
%
\end{equation}
%
\end{dfnsub}
%
\begin{lemsub}\label{blockapp:lem}
%
Let \DE/ and $X$ be a homogeneous Banach space on \T. The following
are equivalent.
\begin{itemize}
\item [$(i)$]$X_E$ has $(u(\tau_f))$, where $\tau_f$ is the topology of
pointwise convergence of the Fourier coefficients:
$$
x_n\hoch{\to}^{\tau_f}0\quad\Longleftrightarrow\quad \forall k\
\widehat{x_n}(k)\to0.
$$
\item [$(ii)$]$E$ enjoys \UP/ in $X$.
\item [$(iii)$]$X_E$ enjoys the property of block unconditionality for
any, or equivalently for some, \as/ of multipliers $\{T_k\}$.
\end{itemize}
\end{lemsub}
%
\dem
%
$(i)\imp(ii)$.
Suppose that $(ii)$ fails: there are
$\eps>0$ and a finite $F$ such that for each finite $G$, there
are $x_G\in B_{X_F}$
and $y_G\in B_{X_{E\setminus G}}$
such that
$$
\osc_{\epsilon\in\sU}\|\epsilon x_G+y_G\|>\eps.
$$
As $B_{X_F}$ is compact, we may suppose $x_G=x$.
As $y_G\hoch{\to}^{\tau_f}0$, $(u(\tau_f))$ fails.
$(ii)\imp(iii)$.
Let $C$ be a uniform bound for $\|T_k\|$.
Let $n\ge1$ and $\eps>0$. Let $F$ be the finite spectrum of
$T_n$. Let $G$ be such that \eqref{block:bloc} holds
for all $f\in B_{X_F}$
and $g\in B_{X_{E\setminus G}}$.
Now there is a term $V$ in de la Vall\'ee-Poussin's
\as/ such that $V|_{X_G}=\Id|_{X_G}$ and
$\|V\|_{\mathscr{L}(X_E)}\le3$.
As $V$ has finite rank, we may choose $m>n$ such that
$\|(\Id -T_m)V\|_{\mathscr{L}(X_E)}=\|V(\Id -T_m)\|_{\mathscr{L}(X_E)}\le\eps$.
Let then $x\in T_nB_{X_E}$ and $y\in(\Id -T_m)B_{X_E}$.
We have
%
\begin{eqnarray*}
%
\|\epsilon x+y\|&\le&\|\epsilon x+(\Id -V)y\|+\eps
\hoch{\le}^{(\text{\ref{block:bloc}})}\|x+(\Id -V)y\|+4(C+1)\eps+\eps\\
&\le&\|x+y\|+(4C+6)\eps.%\\
%
\end{eqnarray*}
$(iii)\imp(i)$ is proved as Lemma \ref{block:lm}
$(ii)\imp(i)$: note that if $y_j\hoch{\to}^{\tau_f}0$,
then $\|Ty_j\|\to0$ for any finite rank multiplier $T$.
%
\eck
%
\vskipa
We may now prove the main result of this section.
%
\begin{thmsub}\label{sbd:thm}
\index{metric unconditional approximation property!for homogeneous Banach spaces}
%
Let \DE/ and $X$ be a homogeneous Banach space on \T. If $X_E$ has
\umap/, then $E$ enjoys \UP/ in $X$. Conversely, if $E$ enjoys \UP/
in $X$ and furthermore $X_E$ has \uap/ and finite cotype, or simply
\lap{1}, then $X_E$ has \umap/. In particular,
\begin{itemize}
\item [$(i)$]For $1 0$ and finite $F\se E$, there is a finite $G\se F$ such
that for $f\in B_{X_F}$ and $g\in B_{X_{E\setminus G}}$
$$
\bigl|\|f+g\|_X-(\|f\|_X^p+\|g\|_X^p)^{1/p}\bigr|\le\eps
$$
\item [$(iii)$] $X_E$ enjoys $m_p(T_k)$ for any, or equivalently for some,
\as/ of multipliers.
\end{itemize}
\end{lemsub}
%
\begin{prpsub}
\index{metric $p$-additive approximation property!for homogeneous Banach spaces}
%
Let \DE/ and $X$ be a homogeneous Banach space.
\begin{itemize}
\item [$(i)$] If $X_E$ has \lpmap/, then $E$ enjoys $\mathscr{M}_p$ in $X$.
\item [$(ii)$] $\SLE{q}$ has \lpmap/ if and only if $p=q=2$.
\item [$(iii)$] \index{metric 1-additive approximation property!for spaces $\SCE$}
$\SCE$ has \lmap{1} if and only if it has \lap{1}
and $E$ enjoys $\mathscr{M}_1$ in $\SC{}$: for all $\eps>0$ and
finite $F\se E$, there is a finite $G\se E$ such that
$$
\forall f\in\SC{F}\ \forall g\in\SC{E\setminus G}\qquad
\|f\|_\infty+\|g\|_\infty\le(1+\eps)\|f+g\|_\infty.
$$
\end{itemize}
\end{prpsub}
%
\dem
$(i)$
Let $\eps>0$. Let $\{T_k\}$ be
an a.s.\ of multipliers that satisfies \eqref{block:def:str} with
$C<1+\eps$. By the
argument of \cite[Lemma 5]{li96}, we may assume that the $T_k$'s have
their range in \PTE/.
Let $n\ge1$ be such that $\bigl(\sum_{k>n}\|\Delta
T_kf\|_X^p\bigr)^{1/p}<\eps$ for $f\in B_{X_F}$. Let $G$ be such that
$T_kg=0$ for $k\le n$ and $g\in X_{E\setminus G}$. Then successively
$$
\Bigl|\Bigl(
\sum_{k\le n}\|\Delta T_k(f+g)\|_X^p\Bigr)^{1/p}-
\Bigl(\sum\|\Delta T_kf\|_X^p\Bigr)^{1/p}\Bigr|
\le\eps,
$$
$$
\Bigl|\Bigl(
\sum_{k> n}\|\Delta T_k(f+g)\|_X^p
\Bigr)^{1/p}-
\Bigl(\sum\|\Delta T_kg\|_X^p\Bigr)^{1/p}\Bigr|
\le\eps,
$$
$$
\Bigl|\Bigl(\sum
\|\Delta T_k(f+g)\|_X^p\Bigr)^{1/p}-
\Bigl(\sum\|\Delta T_kf\|_X^p+\sum\|\Delta T_kg\|_X^p\Bigr)^{1/p}
\Bigr|\le2^{1/p}\eps
$$
and
$$
\bigl|
\|f+g\|_X-(\|f\|_X^p+\|g\|_X^p)^{1/p}
\bigr|\le2\eps(1+2^{1/p}).
$$
$(ii)$
By Corollary \ref{appen:uap}, we necessarily have $p=2$. Furthermore,
if $\SLE{q}$ has \lmap{2}, then by property $\mathscr{M}_2$
$$
\bigl|\|\e_n+\e_m\|_q-\sqrt{2}\bigr|\tol_{m\to\infty}0.
$$
Now
$\|\e_n+\e_m\|_q=\|1+\e_1\|_q$
is constant and differs from $\|1+\e_1\|_2=\sqrt{2}$ unless $q=2$:
otherwise the only case of equality of the norms $\|\cdot\|_q$ and
$\|\cdot\|_2$ occurs for almost everywhere constant functions.
$(iii)$
Use Theorem \ref{block:strong:thm'}.
\eck
\section{\texorpdfstring{Property \umap/ and arithmetical block independence}{Property (umap) and arithmetical block independence}}
\label{sect:umap}
We may now apply the technique used in the
investigation of \umbs/ in order to obtain
arithmetical conditions analogous to \I{n} (see Def.\ \ref{mub:def:ar})
for \umap/. According to Theorem \ref{sbd:thm}, it suffices
to investigate property \UP/ of block unconditionality: we have
to compute an expression of type $\|f+\epsilon g\|_p$, where the spectra
of $f$ and $g$ are far apart and $\epsilon\in\U$. As before, $\U=\T$
(\vs $\U=\D$) is the complex (\vs real) choice of signs.
\subsection{Property of block independence}\label{ss:block}
To this end,
we return to the notation of Computational lemmas
\ref{mub:calcul} and \ref{mub:culcul}. Define
%
\begin{eqnarray}
%
\lefteqn{\nonumber\Psi_r(\epsilon,z)\quad=\quad\Theta_r((\overbrace{1,\dots,1}^j,
\overbrace{\epsilon,\dots,\epsilon}^{m-j}), z)}&&\\
\nonumber&=&\!\int
\biggl|\e_{r_0}(t)+
\sum_{i=1}^jz_i\e_{r_i}(t)
+\epsilon\sum_{i=j+1}^mz_i\e_{r_i}(t)\biggr|^pdm(t)\\
&=&\sum_{\setbox0\hbox{$\scriptstyle\alpha\in\sN^{m}$}\wd0=0pt\box0}\,\,
{\bip{p/2}\alpha}^2\prod|z_i|^{2\alpha_i}
\label{umap:thet}+
\sum_{\scriptstyle\alpha\ne\beta\in\sN^m\atop\scriptstyle\alpha\sim\beta}
%\sum_{A\in\mathscr{R}_q}\sum_{\alpha\ne\beta\in A}
\bip{p/2}\alpha\bip{p/2}\beta
\epsilon^{\mathop{\lower1pt\hbox{$\Sigma$}}_{i>j}\!\alpha_i-\beta_i}
\prod z_i^{\alpha_i}
\overline{z_i}^{\beta_i}.
%
\end{eqnarray}
%
As in Computational lemma \ref{mub:culcul},
we make the following observation:
%
\begin{ldcsub}\label{umap:calcul}
%
Let $\lstp\zeta0m\in\Z^*$ and $\gamma,\delta$ be as in
\eqref{mub:gammadelta}. If the arithmetic relation
\eqref{mub:ArithRel} holds, then the coefficient of the term
$\epsilon^{\lower1pt\hbox{$\Sigma$}_{i>j}\gamma_i-\delta_i} \prod
z_i^{\gamma_i}\overline{z_i}^{\delta_i}$ in \eqref{umap:thet} is
$\bip{p/2}\gamma\bip{p/2}\delta$ and thus independent of $r$. If
$\sum|\zeta_i|\le p$ or $p$ is not an even integer, this coefficient
is nonzero. If $\smp\zeta0j$ is nonzero \(\vs odd\), then this term is
nonconstant in $\epsilon\in\U$.
%
\end{ldcsub}
%
Thus the following arithmetical property shows up. It is
similar to property \I{n}
of almost independence.
%
\begin{dfnsub}\label{arith:def}
%
Let \DE/ and $n\ge1$.
\begin{itemize}
\item [$(i)$]$E$ enjoys the complex \(\vs real\) property \J{n} of block
independence\index{block independent set of integers} if for any
$\zeta\in\Zeta_n^m$ with $\sm\zeta j$ nonzero \(\vs odd\) and given
$\lst pj\in E$, there is a finite $G\se E$ such that
$\zeta_1p_1+\dots+\zeta_mp_m\ne0$ for all $\lstp p{j+1}m\in
E\setminus G$.
\item [$(ii)$]$E$ enjoys exactly complex \(\vs real\) \J{n} if furthermore
it fails complex \(\vs real\) \J{n+1}.
\item [$(iii)$]$E$ enjoys complex \(\vs real\) \J{\infty} if it enjoys
complex \(\vs real\) \J{n} for all $n\ge1$.
\end{itemize}
\end{dfnsub}
%
The complex (\vs real) property \J{n} means precisely the
following. ``For every finite $F\se E$ there is a finite $G\se E$ such
that for any two representations of any $k\in\Z$ as a sum of $n$
elements of $F\cup(E\setminus G)$
$$
\sm pn=k=\sm{p'}n
$$
one necessarily has
$$
\mes{\{j:p_j\in F\}}=\mes{\{j:p'_j\in F\}}\hbox{ in $\Z$ (\vs in $\Z/2\Z$).''}
$$
Thus property \J{n} has, unlike \I{n},
a complex and a real\index{real vs.\ complex}\index{complex vs.\ real} version. Real \J{n} is strictly
weaker than complex \J{n}: see Section \ref{sect:arith}.
Notice that \J{1} is void and $\J{n+1}\imp\J{n}$ in both
complex and real cases. Also $\I{n}\not\imp\J{n}$:
we shall see in the following section that $E=\{0\}\cup\{n^k\}_{k\ge0}$
provides a counterexample.
The property \J{2} of
real block independence appears implicitly in
\cite[Lemma 12]{li96}.\vskipb
\remsub
In spite of the intricate form of this
arithmetical property, \J{n} is the ``simplest'' candidate,
in some sense, that reflects the features of \UP/:
\begin{itemize}
\item [\bloc ]it must hold for a set $E$ if and only if it holds
for a translate $E+k$ of this set: this explains
$\sum\zeta_i=0$ in Definition \ref{arith:def}$(i)$;
\item [\bloc ]as for the property \UP/ of block independence, it must
connect the break\index{break} of $E$ with its tail\index{tail};
\item [\bloc ]Li gives an example of a set $E$ whose pace does not
tend to infinity while $\SCE$ has \lmap{1}.
Thus no property \J{n} should forbid parallelogram
relations of the type $p_2-p_1=p_4-p_3$, where
$p_1,p_2$ are in the break of $E$ and
$p_3,p_4$ in its tail. This
explains the condition that $\sm\zeta j$ be nonzero
(\vs odd) in
Definition \ref{arith:def}$(i)$.
\end{itemize}
We now repeat the argument of Theorem \ref{mub:thm}
to obtain an analogous statement which relates
property \UP/ of Definition \ref{block:block:def}
with our new arithmetical conditions
%
\begin{lemsub}\label{umap:lem}
\index{Fourier block unconditionality}
\index{block independent set of integers}
%
Let \DEE/ and $1\le p<\infty$.
\begin{itemize}
\item [$(i)$]Suppose $p$ is an even integer. Then $E$ enjoys the complex
\(\vs real\) Fourier block unconditionality property \UP/ in
$\SLP{}$ if and only if $E$ enjoys complex \(\vs real\) \J{p/2}.
\item [$(ii)$]If $p$ is not an even integer and $E$ enjoys complex \(\vs
real\) \UP/ in $\SLP{}$, then $E$ enjoys complex \(\vs real\)
\J{\infty}.
\end{itemize}
\end{lemsub}
%
\dem
Let us first prove the necessity of the arithmetical
property and assume $E$ fails \J{n}: then
there are $\lstp\zeta0m\in\Z^*$ with $\sum\zeta_i=0$,
$\sum|\zeta_i|\le2n$ and $\smp\zeta0j$ nonzero (\vs odd);
there are $\lstp{r}0j\in E$ and sequences
$\lstp{r^l}{j+1}m\in E\setminus\{\lstp n1l\}$ such that
$$
\zeta_0r_0+\dots+\zeta_jr_j+\zeta_{j+1}r_{j+1}^l+\dots+\zeta_mr_m^l=0.
$$
Assume $E$ enjoys \UP/ in $\SLP{}$. Then
the oscillation of $\Psi_r$ in \eqref{umap:thet} satisfies
%
\begin{equation}\label{umap:ApplDef}
%
\osc_{\epsilon\in\sU}
\Psi_{r^l}(\epsilon,z)
\tol_{l\to\infty}0
%
\end{equation}
%
for each $z\in D^m$. The argument is now
exactly the same as in Theorem \ref{mub:thm}:
we may assume that
the sequence of functions $\Psi_{r^l}$ converges
in $\mathscr{C}^\infty(\U\times D^m)$ to a function
$\Psi$. Then by \eqref{umap:ApplDef},
$\Psi(\epsilon,z)$ is constant in $\epsilon$
for each $z\in D^m$, and this is impossible by
Computational lemma \ref{umap:calcul}
if $p$ is either not an even integer or
$p\ge 2n$.
Let us now prove the sufficiency of \J{p/2} when
$p$ is an even integer.
First, let $\Alpha_n^{k,l}=
\{\alpha\in \Alpha_n:\alpha_i=0\mbox{ for }k1$ such that either
\eqref{block:bloc} holds for $\eps=0$ or fails
for any $\eps\le C_p$. We thus get
%
\begin{corsub}
%
Let \DE/ and $p$ be an even integer. If $E$ enjoys complex \(\vs
real\) \UP/ in $\SLP{}$, then there is a partition $E=\bigcup E_k$ into
finite sets such that for any coarser partition $E=\bigcup E'_k$
$$
\forall f\in\PTE/ \quad\osc_{\epsilon_k\in\sU}\Bigl\|\sum
\epsilon_k\pi_{E'_{2k}}f\Bigr\|_p=0
$$
%
\end{corsub}
%
Among other consequences, $E=E_1\cup E_2$
where the $\SLP{E_i}$ have a complex (\vs real)
1-unconditional
\fdd/\index{1-unconditional fdd@1-unconditional \fdd/}.\vskipb
\questsub
Is this rigidity proper to translation invariant subspaces of $\SLP{}$,
$p$ an even integer, or generic for all its subspaces
(see
%recent work by Delbaen--Jarchow--Pe\l czy\'nski
\cite{djp})?
\subsection{Main result}
Lemma \ref{umap:lem} and Theorem \ref{sbd:thm} yield the
main result of this section.
%
\begin{thmsub}\label{umap:thm}
\index{metric unconditional approximation property!for spaces $\SLPE$, $p$ even}
%
Let \DE/ and $1\le p<\infty$.
\begin{itemize}
\item [$(i)$]Suppose $p$ is an even integer. Then $\SLPE$ has complex
\(\vs real\) \umap/ if and only if $E$ enjoys complex \(\vs real\)
\J{p/2}.
\item [$(ii)$]If $p$ is not an even integer and $\SLPE$ has complex \(\vs
real\) \umap/, then $E$ enjoys complex \(\vs real\) \J{\infty}.
\end{itemize}
\end{thmsub}
\begin{corsub}\label{umap:thm:cor}\index{metric unconditional approximation property!for $\SCE$ and $\SLPE$, $p\notin2\N$}
%
Let \DE/.
\begin{itemize}
\item [$(i)$]If $\SCE$ has complex \(\vs real\) \umap/, then $E$ enjoys
complex \(\vs real\) \J{\infty}.
\item [$(ii)$] If any $\SLPE$, $p$ not an even integer, has complex \(\vs
real\) \umap/, then all $\SLPE$ with $p$ an even integer have
complex \(\vs real\) \umap/.
\end{itemize}
\end{corsub}
Suppose $p$ is an even integer. Then Section \ref{sect:arith}
gives various examples of sets such that $\SLPE$
has complex or real \umap/. Proposition \ref{comb:grow}
gives a general growth condition that ensures \umap/.
For $X=\SLP{}$, $p$ not an even integer,
and $X=\SC{}$, however, we encounter the same obstacle
as for \umbs/. Section \ref{sect:arith} only gives
sets $E$ such that $X_E$ fails \umap/. Thus, we
have to prove this property by direct means. This yields four
types of examples of sets $E$ such that the space
$\SCE$ --- and thus by \cite[Th.\ 7]{li96} all
$\SLPE$ $(1\le p<\infty)$ as well --- have \umap/.
\begin{itemize}
\item [\bloc]Sets found by Li\index{Li, Daniel} \cite{li96}:
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\index{Meyer, Yves} \cite[VIII]{me72} techniques are
used to construct a Hilbert set\index{Hilbert set}.
\item [\bloc]The sets that satisfy the growth condition of Theorem
\ref{positif:thm};
\item [\bloc]Sequences \DEE/ such that $n_{k+1}/n_k$ is an odd integer:
see Proposition \ref{res:geo}.\vskipb
\end{itemize}
\questsub
We know no example of a set $E$ such that
some $\SLPE$, $p$ not an even integer, has \umap/
while $\SCE$ fails it.\vskipb
There is also a good arithmetical description of the case
where $\{\pi_k\}$ or a subsequence
thereof realises \umap/.
%
\begin{prpsub}\label{umap:prp:fdd}
%
\index{1-unconditional fdd@1-unconditional \fdd/!for spaces $\SLPE$, $p$ even}
\index{metric unconditional fdd@metric unconditional \fdd/}
Let \DEEE/. Consider a partition
$E=\bigcup_{k\ge1}E_k$ into finite sets.
\begin{itemize}
\item [$(i)$]Suppose $p$ is an even integer. The series $\sum\pi_{E_k}$
realises complex \(\vs real\) \umap/ in $\SLPE$ if and only if there
is an $l\ge1$ such that
\begin{equation}
\label{umap:fdd}
\left\{\begin{array}{l}
\lst pm\in E\\
\zeta_1p_1+\dots+\zeta_mp_m=0
\end{array}\right.
\quad\imp\quad
\forall k\ge l\suml_{p_j\in E_k}\zeta_j=0
\ (\mbox{\vs is even})
\end{equation}
for all $\zeta\in\Zeta_{p/2}^m$. Then $\SLPE$ admits the series
$\pi_{\cup_{k \| w_1+{\fam0 i} w_n\|_p=\sqrt{2}.
$$
This is simply due to the fact that the image domain
of the characters on $\D^\infty$ is
too small. Take now
the infinite torus $\T^\infty$
and consider the
set $S=\{s_i\}$ of Steinhaus functions,
\ie the coordinate
functions on $\T^\infty$:
they form again a family of independent
random variables with values uniformly distributed in $\T$.
Then $S$ is clearly a complex
1-\ubs/
\index{1-unconditional basic sequence of characters!on the infinite torus}
in any homogeneous Banach space $X$ on $\T^\infty$.
% $X\in\{\mathscr{C}(\T^\infty),
%{\fam0 L}^p(\T^\infty)\,(1\le p<\infty)\}$.
\subsection{Two notions of approximate probabilistic independence}
\label{ss:proba:two}
As the random variables $\{\e_n\}$ also have their
values uniformly distributed in $\T$,
some sort of approximate independence
should suffice to draw
the same conclusions as in the case of $S$.
A first possibility is to look at the joint
distribution of
$(\e_{p_1},\dots,\e_{p_n})$,
$\lst pn\in E$, and to
ask it to be close to the product of the distributions
of the $\e_{p_i}$. For example,
Pisier\index{Pisier, Gilles}
\cite[Lemma 2.7]{pi83}
gives the following characterisation:
$E$ is a Sidon\index{Sidon set} set if and only if there are a
neighbourhood $V$ of 1
in $\T$ and
$0<\varrho<1$ such that for any finite $F\se E$
\begin{equation}\label{proba:pisier}
m[\e_p\in V:p\in F]\le\varrho^{\smes{F}}.
\end{equation}
Murai \index{Murai, Takafumi}\cite[\S4.2]{mu82}
calls
\DE/ pseudo-independent\index{pseudo-independent set} if for all $\lst An\se\T$
\begin{equation}\label{proba:pi}
m[\e_{p_i}\in A_i:1\le i\le n]
\tol_{
\scriptstyle p_i\in E\atop
\scriptstyle p_i\to\infty}
\prod_{i=1}^n
m[\e_{p_i}\in A_i]=\prod_{i=1}^nm[A_i].
\end{equation}
We have
%His theorem \cite[Lemma 30]{mu82} gives together
%with Proposition \ref{arith:lim}$(iii)$ the following
%
\begin{prpsub}\label{proba:murai}
%
Let \DE/. The following are equivalent.
\begin{itemize}
\item [$(i)$]$E$ is pseudo-independent,
\item [$(ii)$]$E$ enjoys
\I{\infty},
\item [$(iii)$]For every $\eps>0$ and $m\ge1$, there is a finite subset $G\se E$ such
that the Sidon constant of any subset of
$E\setminus G$ with $m$ elements is less than $1+\eps$.
\end{itemize}
\end{prpsub}
%
Note that by Corollary \ref{arith:cor},
\eqref{proba:pi} does not imply \eqref{proba:pisier}.
\dem
$(i)\ssi(ii)$ follows by Proposition \ref{arith:lim}$(iii)$ and
\cite[Lemma 30]{mu82}. $(iii)\imp(ii)$ is true because $(iii)$ is just
what is needed to draw our conclusion in Corollary
\ref{arith:cor}. Let us prove $(i)\imp(iii)$. Let $\eps>0$, $m\ge1$
and $\mathscr{A}$ be a covering of $\T$ with intervals of length
$\eps$. By \eqref{proba:pi}, there is a finite set $G\se E$ such that
for $\lst pm\in E\setminus G$ and $A_i\in\mathscr{A}$ we have
$m[\e_{p_i}\in A_i:A_i\in\mathscr{A}]>0$. But then
$$\Bigl\|\sum a_i\e_{p_i}\Bigr\|_\infty\ge\sum|a_i|\cdot(1-\eps).\eqno{\ecks}$$
\remsub
$(ii)\imp(iii)$ may be proved directly by the
technique of Riesz products\index{Riesz product}:
see \cite[Appendix V, \S1.II]{ks63}.\vskipb
Another possibility is to define some notion of
almost independence. Berkes \cite{be87} introduces
the following notion: let us call a sequence
of random variables $\{X_n\}$ almost
i.i.d.\index{almost i.i.d. sequence}
(independent and
identically distributed) if, after enlarging
the probability space, there is an i.i.d.\ sequence $\{Y_n\}$
such that $\|X_n-Y_n\|_\infty\to0$. We have the straightforward
%
\begin{prpsub}\label{proba:berkes}
%
Let \DEE/. If $E$ is almost i.i.d., then $E$ is a Sidon
set\index{Sidon set!with constant asymptotically 1} with constant
asymptotically 1.
%
\end{prpsub}
%
\dem
%
Let $\{Y_j\}$ be an i.i.d sequence and suppose
$\|\e_{n_j}-Y_j\|_\infty\le\eps$ for $j\ge k$.
Then
$$
\sum_{j\ge k}|a_j|=
\Bigl\|\sum_{j\ge k}a_jY_j\Bigr\|_\infty\le
\Bigl\|\sum_{j\ge k}a_j\e_{n_j}\Bigr\|_\infty+
\eps\sum_{j\ge k}|a_j|
$$
and the unconditionality constant of
$\{\lstf nk\}$ is less than $(1-\eps)^{-1}$.
%
\eck\vskipb
Suppose \DEE/ is such that $n_{k+1}/n_k$
is an integer for all $k$.
In that case, Berkes \cite{be87} proves that $E$ is
almost i.i.d.\ if and only if
$n_{k+1}/n_k\to\infty$.
We thus recover a part of Theorem \ref{positif:thm}.
\questsub
What about the converse in Proposition \ref{proba:berkes}?
\section{Summary of results. Remarks and questions}
\label{sect:resume}
For the convenience of the reader, we now reorder
our results by putting together those which are
relevant to a given class of Banach spaces.
Let us first summarise our arithmetical results on the geometric sequence
$G=\{j^k\}_{k\ge0}$ ($j\in\Z\setminus\{-1,0,1\}$). The number given in the
first (\vs second, third) column is the value $n\ge1$ for which the set in the
corresponding row achieves exactly \I{n} (\vs complex \J{n}, real
\J{n}).\vskipa
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|r||c|c|c|} \hline
$G=\{j^k\}_{k\ge0}$ with $|j|\ge2$&\I{n} &\C-\J{n}&\R-\J{n}\\\hline
$G$, $j>0$ odd\vphantom{\LARGE I} &$|j|$ &$|j|$ &$\infty$\\
$G$, $j>0$ even &$|j|$ &$|j|$ &$|j|$ \\
$G\cup\{0\}$, $j>0$ odd &$|j|$ &$|j|-1$ &$\infty$\\
$G\cup\{0\}$, $j>0$ even &$|j|$ &$|j|-1$ &$|j|-1$ \\
$G$, $G\cup\{0\}$, $j<0$ odd &$|j|-1$&$|j|$ &$\infty$\\
$G$, $G\cup\{0\}$, $j<0$ even &$|j|-1$&$|j|$ &$|j|$ \\
$G\cup-G$, $G\cup-G\cup\{0\}$, $j$ odd&$1$&$1$ &$\infty$\\
$G\cup-G$, $j$ even &$1$ &$1$ &$|j|$ \\
$G\cup-G\cup\{0\}$, $j$ even &$1$ &$1$ &$|j|/2$ \\\hline
\end{tabular}\nopagebreak\smallskip\\\nopagebreak
\refstepcounter{thm}Table \thethm
\end{center}\index{geometric sequences}
\subsection{\texorpdfstring{The case $X={\fam0 L}^p(\T)$ with $p$ an even integer}{The case X=L\^{}p(T) with p an even integer}}
Let $p\ge4$ be an even integer.
We observed the following facts.
\begin{itemize}
\item [\bloc ]Real and complex \umap/ differ among subspaces $\SLPE$ for each $p$:
consider Proposition \ref{res:geo} or $\SLPE$ with $E=\{\pm(p/2)^k\}$.
\item [\bloc ]By Theorem \ref{umap:thm}, $\SLPE$ has complex (\vs real)
\umap/ if so does $\SLE{p+2}$;
\item [\bloc ]The converse is false for any $p$. In the complex case,
$E=\{(p/2)^k\}$ is a counterexample. In the real
case, take $E=\{0\}\cup\{\pm p^k\}$.
\item [\bloc ]Property \umap/ is
not stable under unions with an element: for each $p$,
there is a set $E$
such that $\SLPE$ has complex (\vs real)
\umap/, but $\SLP{E\cup\{0\}}$ does not.
In the complex case, consider $E=\{(p/2)^k\}$.
In the real case, consider
$E=\{\pm(2\lceil p/4\rceil)^k\}$.
\item [\bloc ]If $E$ is a symmetric set and $p\ne2$, then $\SLPE$ fails complex \umap/.
Proposition \ref{arith:sym} gives a criterion for real \umap/.
\end{itemize}
What is the relationship between \umbs/ and complex \umap/? We have by
Proposition \ref{arith:csq}$(i)$ and \ref{umap:prp:fdd}$(i)$
%
\begin{prpsub}\label{resume:mb_um}
%
Let \DEE/ and $n\ge1$.
\begin{itemize}
\item [$(i)$]If $E$ is a \umbs/ in $\SL{4n-2}{}$, then $\SLE{2n}$ has complex
\umap/.
\item [$(ii)$]If $\{\pi_k\}$ realises complex \umap/ in $\SLE{2n}$, then $E$
is a \umbs/ in $\SL{2n}{}$.
\end{itemize}
\end{prpsub}
We also have, by Proposition \ref{comb:thm}$(i)$
%
\begin{prpsub}
%
Let \DE/ and $p\ne2,4$ an even integer. If $\SLPE$ has real \umap/,
then $d^*(E)=0$.
%
\end{prpsub}
Note also this consequence of Propositions
\ref{mub:trans}, \ref{arith:trans}, \ref{proba:murai} and Theorems
\ref{mub:thm}, \ref{umap:thm}
%
\begin{prpsub}
%
Let $\sigma>1$ and $E=\{[\sigma^k]\}$. Then the following properties
are equivalent:
\begin{itemize}
\item [$(i)$]$\sigma$ is transcendental\index{transcendental numbers};
\item [$(ii)$]$\SLPE$ has complex \umap/ for any even integer $p$;
\item [$(iii)$]$E$ is a \umbs/ in any $\SLP{}$, $p$ an even integer;
\item [$(iv)$]$E$ is pseudo-independent.
\item [$(v)$]For every $\eps>0$ and $m\ge1$, there is an $l$ such that for
$\lst km\ge l$ the Sidon constant of
$\{[\sigma^{k_1}],\dots,[\sigma^{k_m}]\}$ is less than $1+\eps$.
\end{itemize}
\end{prpsub}
\subsection{\texorpdfstring{Cases $X={\fam0 L}^p(\T)$ with $p$ not an even
integer and $X=\mathscr C(\T)$}{Cases X=L\^{}p(T) with p
not an even integer and X=C(T)}}
In this section, $X$ denotes either
$\SLP{}$, $p$ not an even integer, or $\SC{}$.
Theorems \ref{mub:thm} and \ref{umap:thm}
only permit us to use the negative results of Section
\ref{sect:arith}: thus, we can just gather
negative results about the functional properties
of $E$.
For example, we know by
Proposition \ref{arith:csq}$(iv)$ that \I{\infty} and \J{\infty} are
stable under union with an element. Nevertheless,
we cannot conclude that the same holds
for \umap/.
The negative results are (by Section \ref{sect:arith}):
\begin{itemize}
\item[\bloc]
for any infinite \DE/,
$X_{E\cup2E}$ fails real \umap/. Thus \umap/
is not stable under unions;
\item[\bloc]
if $E$ is a polynomial sequence (see Section \ref{sect:arith}),
then $E$ is not a
\umbs/ in $X$ and $X_E$ fails real
\umap/;
\item[\bloc]
if $E$ is a symmetric set, then $E$ is not a
\umbs/ in $X$
and $X_E$ fails complex \umap/. Proposition \ref{arith:sym}
gives a criterion for real \umap/;
\item[\bloc]
if $E=\{[\sigma^k]\}$ with
$\sigma>1$ an algebraic number ---~in particular
if $E$ is a
geometric sequence~---, then $E$ is not a
\umbs/ in $X$
and $X_E$ fails complex \umap/.
\end{itemize}
Furthermore, by Proposition \ref{res:geo},
real and complex \umap/ differ in X.
Theorem \ref{positif:thm} is the only but general
positive result on \umbs/ and complex \umap/ in $X$.
Proposition \ref{res:geo} yields further examples
for real \umap/.
What about the sets that satisfy
\I{\infty} or \J{\infty}? We only know that \I{\infty}
does not even ensure Sidonicity by Corollary \ref{arith:cor}.
One might wonder whether for some reasonable class of sets
$E$, $E$ is a finite
union of sets that enjoy \I{\infty} or \J{\infty}.
This is false
even for Sidon sets: for example, let $E$ be the geometric
sequence $\{j^k\}_{k\ge0}$ with $j\in\Z\setminus\{-1,0,1\}$
and suppose $E=E_1\cup\dots\cup E_n$.
Then $E_i=\{j^k\}_{k\in A_i}$, where the $A_i$'s are a
partition of the set of positive integers. But then one
of the $A_i$ contains arbitrarily large $a$ and $b$
such that $|a-b|\le n$. This means that there is an infinite
subset $B\se A_i$
and an $h$, $1\le h\le n$, such that $h+B\se A_i$.
We may apply Proposition \ref{arith:csq}$(vi)$:
$E_i$ enjoys neither \I{j^h+1} nor complex
\J{j^h+1} --- nor real \J{j^h+1}
if furthermore $j$ is even.
Does Proposition \ref{resume:mb_um}$(ii)$ remain true for general
$X$? We do not know this. Suppose however that we know that
$\{\pi_k\}$ realises \umap/ in the following strong manner: for any
$\eps>0$, a tail $\{\pi_k\}_{k\ge l}$ is a $(1+\eps)$-unconditional
a.s.\ in $X_E$.
Then $E$ is trivially a \umbs/ in $X$. In particular, this is the case
if
$$1+\eps_n=
\sup_{\epsilon\in\sU}\|\Id -(1+\epsilon)\pi_n\|_{\mathscr{L}(X)}$$
converges so rapidly to 1 that $\sum\eps_n<\infty$. Indeed,
$$
\sup_{\epsilon_k\in\sU}
\|\pi_{n-1}+\sum_{k\ge n}\epsilon_k\Delta\pi_k\|
\le(1+\eps_n)
\sup_{\epsilon_k\in\sU}
\|\pi_n+\sum_{k>n}\epsilon_k\Delta\pi_k\|.
$$
and thus, for all $f\in\PTE/$,
$$
\sup_{\epsilon_k\in\sU}
\|\pi_lf+\sum_{k>l}\epsilon_k\Delta\pi_kf\|
\le\prod_{k>l}(1+\eps_k)\,\|f\|.
%\tol_{l\to\infty}\|f\|.
$$
%Is there a result corresponding to Proposition \ref{resume:mb_um}$(ii)$? We
%do not know. But suppose that
%$1+\eps_n=\|\Id -(1+\epsilon)\pi_n\|_{\mathscr{L}(X)}$
%converges sufficiently rapidly to 1: suppose
%that not only $\eps_n\to0$ but also $\sum\eps_n<\infty$. As
%$$
%\max_{\epsilon_i\in\sT}
%\biggl\|\sum_{i=l}^k\epsilon_ia_i\e_{n_i}
%+\sum_{i>k}a_i\e_{n_i}\biggr\|_X\le
%(1+\eps_k)\max_{\epsilon_i\in\sT}
%\biggl\|
%\sum_{i=l}^{k-1}\epsilon_ia_i\e_{n_i}
%+\sum_{i\ge k}a_i\e_{n_i}\biggr\|_X,
%$$
%we get
%$$
%\max_{\epsilon_i\in\sT}\Bigl\|\suml_{i\ge l}\epsilon_ia_i\e_{n_i}\Bigr\|_X
%\le\prodl_{i\ge l}(1+\eps_i)\Bigl\|\suml_{i\ge l}a_i\e_{n_i}\Bigr\|_X,
%$$
%and if $\sum\eps_n<\infty$, then
%$(1+\eps_l)(1+\eps_{l+1})\cdots\mathop{\tol}\limits_{l\to\infty}1$, \ie
%$E$ is an \umbs/.
Let us finally state
%
\begin{prpsub}
%
Let \DE/. If $X_E$ has real \umap/, then $d^*(E)=0$.
%
\end{prpsub}
\subsection{Questions}
The following questions remain open:
\vskipa{\bf Combinatorics } Regarding Proposition \ref{comb:thm}$(i)$,
is there a set $E$ enjoying \J{2} with positive maximal density, or
even with a uniformly bounded pace? Furthermore, may a set $E$ with
positive maximal density admit a partition $E=\bigcup E_i$ in finite sets
such that all $E_i+E_j$, $i\le j$, are pairwise disjoint? Then
$\SLE{4}$ would admit a 1-unconditional
\fdd/\index{1-unconditional fdd@1-unconditional \fdd/!for $\SLE{4}$}
by Proposition
\ref{umap:prp:fdd}$(i)$.
\vskipa{\bf Functional analysis } Let $X\in\{\SL{1}{},\SC{}\}$ and
consider Theorem \ref{sbd:thm}. Is \UP/ sufficient for $X_E$ to share
\umap/? Is there a set \DE/ such that some space $\SLPE$, $p$ not an
even integer, has \umap/, while $\SCE$ fails it?
\vskipa{\bf Harmonic analysis } Is there a Sidon set \DEE/ of constant
asymptotically 1 such that $n_{k+1}/n_k$ is uniformly bounded?
%Is it possible that for any $\eps>0$ there are $n_3\le qn_2\le q^2n_1$
%such that
%$$
%\forall a_1,a_2,a_3\quad
%(1+\eps)\|a_1\e_{n_1}+a_2\e_{n_2}+a_3\e_{n_3}\|_\infty
%\ge|a_1|+|a_2|+|a_3|\ ?$$
What about the case $E=[\sigma^k]$ for a
transcendental\index{transcendental numbers}
$\sigma>1$? If $E$ enjoys \I{\infty}, is $E$ a \umbs/
in $\SLP{}$ $(1\le p<\infty)$? What about \J{\infty}?
%
%
%
%
\vfill\pagebreak
\phantomsection
\addcontentsline{toc}{section}{Bibliography}
\def\cprime{$'$}\def\bibmath{disponible \`a la biblioth\`eque de
math\'ematiques}\def\bibirem{disponible \`a la biblioth\`eque de
l'IREM}\def\BUL{disponible \`a la BU Lettres}\def\BUS{disponible \`a la BU
Sciences}\ifx\iflanguage\undefined\def\iflanguage#1#2#3{#3}\fi
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%\baselineskip=13pt minus 1pt
\vfill\pagebreak
\section*{Index of notation}
\addcontentsline{toc}{section}{Index of notation}
\halign{\hfil#&\quad#\hfil\cr
%symbols
$\mes{B}$&cardinal of $B$\cr
$X_E$&space of $X$-functions with spectrum in $E$\cr
$\widehat{f}$&Fourier transform of $f$: $\widehat{f}(n)=\int
f(t)\e_{-n}(t)dm(t)$\cr
$x\choose\alpha$&multinomial number, \S\ref{ss:almost}\cr
$\XE$&pairing of the arithmetical relation $\zeta$ against
the spectrum $E$, \S\ref{ss:mubs:gen}\cr
$u_n\preccurlyeq v_n$&$|u_n|$ is bounded by $C|v_n|$ for some $C$\cr
\noalign{\vskip4pt minus 0.5pt}
%ciphers
1-\ubs/&1-unconditional basic sequence of characters, Def.\
\ref{mub:def}$(i)$\cr \noalign{\vskip4pt minus 0.5pt}
%a
$A(\T)$&disc algebra $\SC{\sN}$\cr
$\Alpha_n,\Alpha_n^m$&sets of multi-indices viewed as arithmetic
relations, \S\ref{ss:isom}\cr
a.s.&approximating sequence, Def.\ \ref{block:def}\cr\noalign{\vskip4pt minus 0.5pt}
%b
$B_X$&unit ball of the Banach space $X$\cr\noalign{\vskip4pt minus 0.5pt}
%c
$\SC{}$&space of continuous functions on $\T$\cr
\noalign{\vskip4pt minus 0.5pt}
%d
$\D$&set of real signs $\{-1,1\}$\cr
$\Delta T_k$&difference sequence of the $T_k$: $\Delta
T_k=T_k-T_{k-1}$ ($T_0=0$)\cr\noalign{\vskip4pt minus 0.5pt}
%e
$\e_n$&character of $\T$: $\e_n(z)=z^n$ for $z\in\T$, $n\in\Z$\cr
\noalign{\vskip4pt minus 0.5pt}
%f
\fdd/&finite dimensional decomposition, Def.\ \ref{block:def}\cr\noalign{\vskip4pt minus 0.5pt}
%h
$H^1(\T)$&Hardy space $\SL{1}{\sN}$\cr\noalign{\vskip4pt minus 0.5pt}
%i
\I{n}&arithmetical property of almost independence, Def.\
\ref{mub:def:ar}\cr
$\Id$&identity\cr
i.i.d.&independent identically distributed, \S\ref{ss:proba:two}\cr\noalign{\vskip4pt minus 0.5pt}
%j
\J{n}&arithmetical property of block independence, Def.\
\ref{arith:def}\cr \noalign{\vskip4pt minus 0.5pt}
%l
$\mathscr{L}(X)$&space of bounded linear operators on the Banach space $X$\cr
$\SLP{}$&Lebesgue space of $p$-integrable functions on $\T$\cr
\lpap/&$p$-additive approximation property, Def.\ \ref{str:def}\cr
\lpmap/&metric $p$-additive approximation property, Def.\
\ref{str:def}\cr
$\Lambda(p)$&Rudin's class of lacunary sets, Def.\ \ref{mub:sido}\cr\noalign{\vskip4pt minus 0.5pt}
%m
$\mathscr{M}_p$&functional property of Fourier block $p$-additivity,
Lemma \ref{sbd:lpmap:lem}$(ii)$\cr
$\mathscr{M}(\T)$&space of Radon measures on $\T$\cr
$m[A]$&measure of $A\se\T$\cr
$(m_p(\tau))$&functional property of $\tau$-$p$-additivity,
Def.\ \ref{block:strong:dfn}$(i)$\cr
$(m_p(T_k))$&functional property of commuting block $p$-additivity,
Def.\ \ref{block:strong:dfn}$(ii)$\cr\noalign{\vskip4pt minus 0.5pt}
%o
$\osc f$&oscillation of $f$\cr\noalign{\vskip4pt minus 0.5pt}
%p
$\PT{}$&space of trigonometric polynomials on $\T$\cr
$\pi_j$&projection of $X_E$, $E=\{n_k\}$, onto $X_{\{\lst n j\}}$\cr
$\pi_F$&projection of $X_E$ onto $X_F$\cr\noalign{\vskip4pt minus 0.5pt}
%s
$\U$&real ($\U=\D$) or complex ($\U=\T$) choice of signs\cr
\noalign{\vskip4pt minus 0.5pt}
%t
$(\T,dm)$&unit circle in $\C$ with its normalised Haar measure\cr
$\tau_f$&topology of pointwise convergence of the Fourier
coefficients, Lemma\ \ref{blockapp:lem}$(i)$\cr\noalign{\vskip4pt minus 0.5pt}
%u
\UP/&functional property of Fourier block unconditionality, Def.\
\ref{block:block:def}\cr
$(u(\tau))$&functional property of $\tau$-unconditionality, Def.\
\ref{block:def:u}$(i)$\cr
$(u(T_k))$&functional property of commuting block
unconditionality, Def.\ \ref{block:def:u}$(ii)$\cr
\uap/&unconditional approximation property, Def.\
\ref{block:def}\cr
\ubs/&unconditional basic sequence, Def.\ \ref{mub:def}\cr
\umap/&metric unconditional approximation property, Def.\
\ref{block:def}\cr
\umbs/&metric unconditional basic sequence, Def.\
\ref{mub:def}\cr \noalign{\vskip4pt minus 0.5pt}
%z
$\Zeta^m,\Zeta_n^m$&sets of multi-indices viewed as arithmetic relations,
\S\ref{ss:isom}\cr
}
\addcontentsline{toc}{section}{Index}
\begin{flushleft}
\begin{theindex}
\item 1-unconditional basic sequence of characters, \hyperpage{14}
\subitem in $\SC{}$ and $\SLP{}$, $p\notin2\N$, \hyperpage{15},
\hyperpage{17}
\subitem in spaces $\SLP{}$, $p$ even, \hyperpage{15},
\hyperpage{38}
\subitem on the Cantor group, \hyperpage{40}
\subitem on the infinite torus, \hyperpage{41}
\item 1-unconditional \fdd/, \hyperpage{34}
\subitem for $\SLE{4}$, \hyperpage{43}
\subitem for spaces $\SLPE$, $p$ even, \hyperpage{35}
\indexspace
\item almost i.i.d. sequence, \hyperpage{41}
\item almost independence, \hyperpage{19}
\item approximating sequence, \hyperpage{22}
\item approximation property, \hyperpage{22}
\item arithmetical relation, \hyperpage{15}, \hyperpage{17, 18}
\indexspace
\item Binet, J. P. M., \hyperpage{37}
\item birelation, \hyperpage{15}
\item Bishop, Errett A., \hyperpage{30}
\item Blei, Ron C., \hyperpage{27}
\item block independent set of integers, \hyperpage{33}
\item boundedly complete approximating sequence, \hyperpage{25}
\item Bourgain, Jean, \hyperpage{12}
\item break, \hyperpage{23}, \hyperpage{33}
\indexspace
\item Cantor group, \hyperpage{14}, \hyperpage{17}, \hyperpage{40}
\item Carleson, Lennart, \hyperpage{30}
\item Casazza, Peter G., \hyperpage{22}
\item commuting block unconditionality, \hyperpage{22}
\item complex vs.\ real, \hyperpage{14}, \hyperpage{17},
\hyperpage{33}, \hyperpage{40}
\item cotype, \hyperpage{25}
\indexspace
\item Daugavet property, \hyperpage{30}
\indexspace
\item equimeasurability, \hyperpage{12}
\item Erd\H{o}s, Paul, \hyperpage{40}
\item Euler's conjecture, \hyperpage{21}, \hyperpage{37}
\item Euler, Leonhard, \hyperpage{37}
\item exponential growth, \hyperpage{40}
\indexspace
\item Fibonacci sequence, \hyperpage{37}
\item finite-dimensional Schauder decomposition, \hyperpage{22}
\item Forelli, Frank, \hyperpage{11}
\item Fourier block unconditionality, \hyperpage{30}, \hyperpage{33}
\item Fr\'enicle de Bessy, Bernard, \hyperpage{21}
\indexspace
\item geometric sequences, \hyperpage{20}, \hyperpage{36, 37},
\hyperpage{42}
\item Godefroy, Gilles, \hyperpage{23}, \hyperpage{28}
\indexspace
\item Hadamard set, \hyperpage{39}
\item Hilbert set, \hyperpage{27}, \hyperpage{34}
\item Hindman, Neil, \hyperpage{39}
\item homogeneous Banach space, \hyperpage{13}
\indexspace
\item independent set of integers, \hyperpage{15}
\item infinite difference set, \hyperpage{40}
\item isometries on ${\fam0 L}^p$, \hyperpage{17}
\indexspace
\item Kadets, Vladimir M., \hyperpage{30}
\item Kalton, Nigel J., \hyperpage{11}, \hyperpage{22, 23},
\hyperpage{25}, \hyperpage{28}
\item Kazhdan, David A., \hyperpage{39}
\indexspace
\item \EL{p} set, \hyperpage{15}, \hyperpage{35}
\subitem constant, \hyperpage{15}
\item Li, Daniel, \hyperpage{28}, \hyperpage{34}, \hyperpage{39}
\item Littlewood--Paley partition, \hyperpage{29}
\item Lust-Piquard, Fran\c coise, \hyperpage{30}
\indexspace
\item maximal density, \hyperpage{39}
\subitem of block independent sets, \hyperpage{40}
\subitem of independent sets, \hyperpage{39}
\item metric $p$-additive approximation property, \hyperpage{25},
\hyperpage{27}
\subitem for homogeneous Banach spaces, \hyperpage{31}
\subitem for subspaces of ${\fam0 L}^p$, \hyperpage{28}
\item metric 1-additive approximation property
\subitem for spaces $\SCE$, \hyperpage{32}, \hyperpage{38}
\subitem for subspaces of ${\fam0 L}^1$, \hyperpage{28}
\item metric unconditional approximation property, \hyperpage{22, 23}
\subitem for $\SCE$ and $\SLPE$, $p\notin2\N$, \hyperpage{34}
\subitem for homogeneous Banach spaces, \hyperpage{31},
\hyperpage{38}
\subitem for spaces $\SLPE$, $p$ even, \hyperpage{34},
\hyperpage{38}
\subitem on the Cantor group, \hyperpage{40}
\item metric unconditional basic sequence, \hyperpage{14},
\hyperpage{19}, \hyperpage{38}
\item metric unconditional \fdd/, \hyperpage{22}, \hyperpage{35}
\item Meyer, Yves, \hyperpage{34}
\item Murai, Takafumi, \hyperpage{20}, \hyperpage{41}
\indexspace
\item oscillation, \hyperpage{13}
\indexspace
\item $p$-additive approximation property, \hyperpage{25}
\subitem for spaces $\SCE$, \hyperpage{27}
\subitem for spaces $\SLPE$, \hyperpage{27}
\item Pe\l czy\'nski, Aleksander, \hyperpage{22}
\item Pisier, Gilles, \hyperpage{41}
\item Plotkin, A. I., \hyperpage{11}
\item polynomial growth, \hyperpage{40}
\item polynomial sequences, \hyperpage{21}, \hyperpage{37}
\subitem biquadrates, \hyperpage{21}, \hyperpage{37}
\subitem cubes, \hyperpage{21}, \hyperpage{37}
\subitem squares, \hyperpage{21}, \hyperpage{37}
\item pseudo-independent set, \hyperpage{41}
\indexspace
\item Rademacher functions, \hyperpage{40}
\item Ramanujan, Srinivasa, \hyperpage{37}
\item real vs.\ complex, \hyperpage{14}, \hyperpage{17},
\hyperpage{33}, \hyperpage{40}
\item relative multipliers, \hyperpage{14}
\subitem interpolation, \hyperpage{14}
\item renormings, \hyperpage{12}
\item Riesz product, \hyperpage{39}, \hyperpage{41}
\item Riesz set, \hyperpage{29}
\item Rosenblatt, Murray, \hyperpage{12}
\item Rosenthal set, \hyperpage{29}
\item Rosenthal, Haskell Paul, \hyperpage{11, 12}
\item Rudin, Walter, \hyperpage{15}, \hyperpage{30}
\indexspace
\item Schur property, 1-strong, \hyperpage{28}
\item semi-Riesz set, \hyperpage{30}
\item shrinking approximating sequence, \hyperpage{25}
\item Sidon set, \hyperpage{15}, \hyperpage{19}, \hyperpage{27},
\hyperpage{41}
\subitem constant, \hyperpage{15}, \hyperpage{39}
\subitem with constant asymptotically 1, \hyperpage{19},
\hyperpage{38}, \hyperpage{41}
\item smoothness, \hyperpage{12}
\item Stein, Elias, \hyperpage{30}
\item strong mixing, \hyperpage{12}
\item sup-norm-partitioned sets, \hyperpage{27}, \hyperpage{30}
\item superexponential growth, \hyperpage{40}
\item superpolynomial growth, \hyperpage{40}
\item symmetric sets, \hyperpage{36, 37}
\indexspace
\item tail, \hyperpage{23}, \hyperpage{33}
\item $\tau$-unconditionality, \hyperpage{22}
\item transcendental numbers, \hyperpage{20}, \hyperpage{36},
\hyperpage{42, 43}
\indexspace
\item unconditional approximation property, \hyperpage{22}
\subitem for spaces $\SCE$, \hyperpage{29}
\subitem for spaces $\SLE{1}$, \hyperpage{29}
\item unconditional basic sequence of characters, \hyperpage{14}
\item unconditional \fdd/, \hyperpage{22}
\item unconditional skipped blocking decompositions, \hyperpage{23}
\item unconditionality constant, \hyperpage{14}
\subitem in $\SL{4}{}$, \hyperpage{16}
\indexspace
\item Werner, Dirk, \hyperpage{25}, \hyperpage{28}, \hyperpage{30}
\item Wojtaszczyk, Przemys\l aw, \hyperpage{22}
\end{theindex}
\end{flushleft}
\end{document}