% typeset using latex \documentstyle[12pt]{article} \def\Bbb#1{{\hbox{\bf #1}}} \def\N{\Bbb N} \def\R{\Bbb R} \def\C{\Bbb C} \def\Z{\Bbb Z} \def\Con{${\cal C}$} \def\M{${\cal M}$} \def\T{\Bbb T} \begin{document} \title{On the Distribution of Sidon Series} \author{Nakhl\'e H.\ Asmar\thanks{Supported by NSF grant DMS 9102044}\ \ and Stephen Montgomery-Smith\thanks{Supported by NSF grant DMS 9001796} \\ University of Missouri--Columbia \\ Columbia, MO 65211 \\ U.\ S.\ A. \date{}} \maketitle \begin{abstract} Let $B$ denote an arbitrary Banach space, $G$\ a compact abelian group with Haar measure $\mu$\ and dual group $\Gamma$. Let $E$ be a Sidon subset of $\Gamma$ with Sidon constant $S(E)$. Let $r_n$ denote the $n$-th Rademacher function on $[0 , 1]$. We show that there is a constant $c$, depending only on $S(E)$, such that, for all $\alpha > 0$: \begin{eqnarray*} c^{-1}P\left[\left\| \sum_{n=1}^Na_nr_n\right\| \geq c\alpha \right] & \leq & \mu\left[ \left\| \sum_{n=1}^Na_n\gamma_n\right\|\geq \alpha \right] \\ & & \leq \ c\,P\left[\left\| \sum_{n=1}^Na_nr_n\right\| \geq c^{-1}\alpha \right] \end{eqnarray*} where $a_1$, $\ldots$ , $a_N$ are arbitrary elements of $B$, and $\gamma_1$ , $\ldots$ , $\gamma_N$ are arbitrary elements of $E$. We prove a similar result for Sidon subsets of dual objects of compact groups, and apply our results to obtain new lower bounds for the distribution functions of scalar-valued Sidon series. We also note that either one of the above inequalities, even in the scalar case, characterizes Sidon sets. \end{abstract} \bigskip \bigskip \bigskip \bigskip \bigskip AMS Subject Classification (1980): 43A46, 43A15, 46E40, 43A77 \pagebreak \section{Introduction} Suppose that $G$ is a compact abelian group with dual group $\Gamma$. Denote the normalized Haar measure on $G$ by $\mu$. Let ${\cal C}$($G$) be the Banach space of continuous complex-valued functions on $G$. If $S$ $\subset \Gamma$ , a function $f \in L^1(G)$ is called $S-spectral$ whenever $\hat{f}$ is supported in $S$, where here and throughout the paper $\hat{}$ denotes taking the Fourier transform. The collection of $S$-spectral functions that belong to a class of functions ${\cal W}$ will be denoted by ${\cal W}_S$. \newtheorem{defsidon}{Definition}[section] \begin{defsidon} A subset $E$ of $\Gamma$ is called a Sidon set if there is a constant $c > 0$, depending only on $E$, such that \begin{equation} \sum_{\gamma \in \Gamma} \left|\hat{f}(\gamma )\right| \leq c\left\| f\right\|_{\infty} \label{eq1} \end{equation} for every $f \in {\cal C}_E(G)$. The smallest constant c such that (\ref{eq1}\ ) holds is denoted by S(E) and is called the constant of sidonicity of E, or the Sidon constant of E. \label{defsidon} \end{defsidon} If $E = \{ \gamma_j\} \subset \Gamma$ is a Sidon set and $\{ a_j\}$ is a sequence in a Banach space $B$, then the formal series $\sum a_j\gamma_j$ will be referred to as a {\em B-valued Sidon series}. The norm on a given Banach space $B$ will be denoted by $\| \cdot \|$, or, sometimes, by $\| \cdot \|_B$. \\ It is well-know that Sidon series share many common properties with Rademacher series. The following theorem of Pisier illustrates this fact and will serve as a crucial tool in our proofs. \newtheorem{th:Pis1}[defsidon]{Theorem} \begin{th:Pis1}[{\cite[Th\'eor\eme 2.1]{bib:Pi1}}] Suppose that $E = \{ \gamma_n \} \subset \Gamma$ is a Sidon set, that $B$ is a Banach space, and that $a_1, \ldots, a_N \in B$. There is a constant $c_1$, depending only on the Sidon constant $S(E)$, such that, for every $p \in [1, \infty [$, we have \begin{equation} c_1^{-1}\left( {\bf E}\left\| \sum_{n=1}^N a_nr_n \right\|^p\right)^{\frac{1}{p}} \leq \left( \int_G \left\| \sum_{n=1}^N a_n \gamma_n \right\|^p d\mu \right)^{\frac{1}{p}} \\ \leq c_1 \left( {\bf E}\left\| \sum_{n=1}^N a_nr_n \right\|^p\right)^{\frac{1}{p}}. \label{ineq:Pis} \end{equation} \label{th:Pis1} \end{th:Pis1} In view of this similarity between Sidon series and Rademacher series, it is natural to ask how the distribution function of a Sidon series compares to the distribution function of a Rademacher series. Our main result provides an answer to this question. \newtheorem{th:main}[defsidon]{Theorem} \begin{th:main} Suppose that $E = \{ \gamma_n\} \subset \Gamma$ is a Sidon set, and let $B$ denote an arbitrary Banach space. There is a constant $c > 0$ that depends only on the Sidon constant $S(E)$, such that for all $a_1, \ldots , a_N \in B$, and all $\alpha > 0$, we have \begin{equation} c^{-1}P\left[ \left\| \sum_{n=1}^N a_nr_n\right\| \geq c\alpha \right] \leq \mu\left[ \left\| \sum_{n=1}^N a_n\gamma_n\right\| \geq \alpha \right] \leq cP\left[ \left\| \sum_{n=1}^N a_nr_n\right\| \geq c^{-1}\alpha \right]. \label{ineq:main} \end{equation} \label{th:main} \end{th:main} Thus the distribution functions of Sidon and Rademacher series are equivalent. Our proof of this result combines well-known properties of Sidon series with Lemma \ref{lem:main} below. This Lemma provides sufficient conditions for the equivalence of distribution functions. It applies as well in the setting of noncommutative groups yielding an analogue of Theorem \ref{th:main}. Using the estimates of \cite{bib:Mo1}, we obtain sharp lower bound estimates on the distribution of scalar-valued Sidon series on compact abelian groups. We should note that the topological implication of our main result is much easier to show, as is done in \cite{bib:Pi1}, that is, the measure topology on the spaces of Rademacher series and the Sidon series are equivalent. \section{A principle for the equivalence of distribution functions} All random variables are defined on some probability space $(\Omega , {\cal M}, dP)$. We denote the set of positive integers by $\N$, the set of integers by $\Z$, and the circle group by $\T$. All other notation is as in Section 1. We start with a couple of preliminary lemmas. \newtheorem{lem:1}{Lemma}[section] \begin{lem:1} Suppose that $f_1 , f_2, \ldots, f_N$ are independent identically distributed random variables , and let $f$ be a function with the same distribution as the $f_j$'s such that $$P[ |f| \geq \alpha ] \geq \frac{\theta}{N}$$ where $\alpha$ and $\theta$ are positive numbers. Then $$P[ \sup_{1\leq j\leq N}|f_j| \geq \alpha ] \geq \frac{\theta}{1 + \theta}.$$ \label{lem:1} \end{lem:1} {\bf Proof.} We first show that, for $\theta > 0$ and $N \in \N$, we have \begin{equation} (1 - \frac{\theta }{N})^N \leq \frac{1}{(1 + \theta )}. \label{2.2} \end{equation} This follows from the inequalities $$(1 - \frac{\theta }{N})(1 + \frac{\theta }{N}) \leq 1$$ and $$(1 + \frac{\theta }{N})^N \geq 1 + \theta .$$ Using (\ref{2.2}\ ) and independence, we get: \begin{eqnarray*} P[\sup_j |f_j| \geq \alpha ] & = & 1 - P[\sup_j |f_j| < \alpha ] \\ & = & 1 -\prod_j P[|f_j| < \alpha ] \\ & \geq & 1 - (1 -\frac{\theta}{N})^N \\ & \geq & 1 - \frac{1}{(1 + \theta )} \\ & = & \frac{\theta}{(1 + \theta )}. \end{eqnarray*} \newtheorem{lem:2}[lem:1]{Lemma} \begin{lem:2} Suppose that $f_1 , f_2, \ldots, f_N$ are independent identically distributed random variables , and let $f$ be a function with the same distribution as the $f_j$'s such that $$P[ |f| \geq \alpha ] \leq \frac{\theta}{N}$$ where $\alpha$ and $\theta$ are positive numbers. Then, we have \begin{equation} P[ \sup_j|f_j| \geq \alpha ] \leq \theta . \label{ineq:lem2} \end{equation} \label{lem:2} \end{lem:2} {\bf Proof.} We start with the inequality \begin{eqnarray*} (1 - x)^n \geq 1 - nx & & ( 0\le x \le 1), \end{eqnarray*} which can be easily proved by using induction and the inequality \begin{eqnarray*} (1 - x)(1 - y) \geq 1 - x - y & & (x , y \geq 0). \end{eqnarray*} Hence, for all $0\le \theta \le N$, $N\in\N$, we have \begin{eqnarray} (1 - \frac{\theta}{N})^N \geq 1 - \theta & & (0\le \theta \le N, N\in\N ). \label{ineq:prf.lem2.1} \end{eqnarray} The proof of (\ref{ineq:lem2}) is now straightforward, using (\ref{ineq:prf.lem2.1}) and independence: \begin{eqnarray*} P[\sup_j |f_j| \geq \alpha ] & = & 1 - P[\sup_j |f_j| < \alpha ] \\ & = & 1 -\prod_j P[|f_j| < \alpha ] \\ & = & 1 - \prod_j(1- P[ |f_j|\geq\alpha ] ) \\ & \leq & 1 - (1 - \frac{\theta}{N})^N \\ & \leq & \theta . \end{eqnarray*} Before stating our main Lemma, we recall two well-known inequalities. Let $X$ denote a random variable on a probability space $(\Omega, {\cal M}, dP)$, then, for all $y > 0$, we have: \begin{eqnarray} P[|X| \geq y{\bf E}|X|] \leq y^{-1} & & (Chebychev's\ Inequality); \label{ineq:Cheb} \end{eqnarray} and, for $0 < y < 1$, we have \begin{eqnarray} P[|X| \geq y{\bf E}|X|] \geq (1 - y)^2\frac{{\bf E}^2|X|}{{\bf E}|X|^2} & & (Paley-Zygmund\ Inequality). \label{ineq:PZ} \end{eqnarray} See \cite[Ineq. II, p. 8]{bib:Ka}. The statement of our main Lemma now follows. \\ \newtheorem{lem:main}[lem:1]{Lemma} \begin{lem:main} Suppose that $X$ and $Y$ are two Banach valued random variables that are not identically zero, and suppose that $\{ X_n\}$ and $\{ Y_n\}$ are two sequences of independent Banach valued random variables such that $X_n$ is identically distributed with $X$, and $Y_n$ is identically distributed with $Y$. Suppose that there are constants $c_1$ and $c_2$ such that, for all positive integers $N$, we have: \begin{equation} c_1^{-1} \left\| \sup_{1 \leq j \leq N}\left\| X_j\right\| \right\|_1 \leq \left\| \sup_{1 \leq j \leq N}\left\| Y_j\right\| \right\|_1 \leq c_1\left\| \sup_{1 \leq j \leq N}\left\| X_j\right\| \right\|_1 ; \label{ineq:lem.main.1} \end{equation} \begin{equation} \frac{\left\| \sup_{1 \leq j \leq N}\left\| X_j\right\| \right\|^2_1} {\left\| \sup_{1 \leq j \leq N}\left\| X_j\right\| \right\|^2_2} \geq c_2 ; \label{ineq:lem.main.2} \end{equation} and \begin{equation} \frac{\left\| \sup_{1 \leq j \leq N}\left\| Y_j\right\| \right\|^2_1} {\left\| \sup_{1 \leq j \leq N}\left\| Y_j\right\| \right\|^2_2} \geq c_2 . \label{ineq:lem.main.3} \end{equation} Then there is a constant $c$, depending only on $c_1$ and $c_2$, such that, for all $\alpha > 0$, we have: \begin{equation} c^{-1}P\left[ \left\| X\right\| \geq c\alpha \right] \leq P\left[ \left\| Y\right\| \geq \alpha \right] \leq cP\left[ \left\| X\right\| \geq c^{-1}\alpha \right]. \label{ineq:lem.main.4} \end{equation} \label{lem:main} \end{lem:main} {\bf Proof.} We start with the second inequality in (\ref{ineq:lem.main.4}). Given an arbitrary $\alpha = \alpha_1 > 0$ with \begin{equation} 0 < P\left[\left\|Y\right\| \geq \alpha_1\right], \label{eq:1} \end{equation} choose $\nu$ to be the smallest positive integer satisfying: \begin{equation} \frac{1}{2\nu}\leq P\left[\left\|Y\right\| \geq \alpha_1\right] \leq \frac{1}{\nu}. \label{eq:2} \end{equation} From Lemma \ref{lem:1}, it follows that \begin{equation} \frac{1}{3} \leq P\left[\sup_{1\leq j\leq \nu} \left\|Y_j\right\| \geq \alpha_1\right]. \label{eq:3} \end{equation} Chebychev's Inequality (\ref{ineq:Cheb}), and (\ref{eq:3}) imply that \begin{equation} \frac{1}{3} \leq \frac{1}{\alpha_1}\left\|\sup_{1\leq j\leq \nu} \left\|Y_j\right\|\right\|_1 . \label{eq:4} \end{equation} From (\ref{ineq:lem.main.1}) and (\ref{eq:4}), we have \begin{equation} \frac{1}{3} \leq \frac{c_1}{\alpha_1}\left\|\sup_{1\leq j\leq \nu} \left\|X_j\right\|\right\|_1 . \label{eq:5} \end{equation} Hence, for any $\alpha_2 > 0$, (\ref{eq:5}) implies that \begin{equation} \frac{\alpha_1}{3 \alpha_2 c_1} \leq \frac{1}{\alpha_2}\left\|\sup_{1\leq j\leq \nu} \left\|X_j\right\|\right\|_1. \label{eq:6} \end{equation} In particular, if $\alpha_2 = \frac{\alpha_1}{6c_1}$, we get from (\ref{eq:6}) \begin{equation} 2\leq \frac{1}{\alpha_2}\left\|\sup_{1\leq j\leq \nu} \left\|X_j\right\|\right\|_1. \label{eq:7} \end{equation} Now we go back to the Paley--Zygmund Inequality (\ref{ineq:PZ}) and apply it to \begin{eqnarray*} \sup_{1\leq j\leq \nu}\left\| X_j\right\| & {\rm with} & y=\frac{\alpha_2}{\left\| \sup_{1\leq j\leq \nu}\left\| X_j\right\| \right\|_1} . \end{eqnarray*} Taking into account (\ref{ineq:lem.main.2}) and noticing from (\ref{eq:7}) that $y\leq \frac{1}{2}$, we get \begin{eqnarray} \frac{1}{4}c_2 & \leq & P\left[\sup_{1\leq j\leq \nu} \left\| X_j\right\| \geq \alpha_2\right] \nonumber \\ & = & P\left[\sup_{1\leq j\leq \nu} \left\| X_j\right\| \geq d\alpha_1\right] \label{eq:8} \end{eqnarray} % where $d = \frac{1}{6c_1}$. Lemma \ref{lem:2} , (\ref{eq:2}), and (\ref{eq:8}) imply that \begin{eqnarray} P\left[\left\| X\right\| \geq d\alpha_1 \right] & \geq & \frac{c_2}{4\nu} \\ & \geq & \frac{c_2}{4}P\left[\left\| Y \right\| \geq \alpha_1\right]. \label{eq:9} \end{eqnarray} % Take $c^{-1} = \min (d , \frac{c_2}{4})$, then (\ref{eq:9}) shows that \begin{equation} P\left[\left\| X\right\| \geq c^{-1}\alpha_1 \right] \geq c^{-1}P\left[\left\| Y\right\| \geq \alpha_1 \right]. \label{eq:10} \end{equation} % Note that (\ref{eq:10}) holds for all $\alpha_1$ for which (\ref{eq:1}) is true. For all other values of $\alpha_1$, inequality (\ref{eq:10}) holds trivially. Thus (\ref{eq:10}) holds for all $\alpha_1 > 0$. Now we repeat the proof with $X$ and $Y$ interchanged. From (\ref{eq:10}) we get: \begin{equation} P\left[\left\| Y\right\| \geq c^{-1}\alpha_1 \right] \geq c^{-1}P\left[\left\| X\right\| \geq \alpha_1 \right], \label{eq:11} \end{equation} % for all $\alpha_1 >0$. Combining (\ref{eq:10}) and (\ref{eq:11}), we obtain \begin{equation} c^{-1}P\left[\left\| X\right\| \geq \alpha_1 \right] \leq P\left[\left\| Y\right\| \geq c^{-1}\alpha_1 \right] \leq cP\left[\left\| X\right\| \geq c^{-2}\alpha_1 \right] \label{eq:12} \end{equation} % for all $\alpha_1 > 0$. Equivalently, we have \begin{equation} c^{-1}P\left[\left\| X\right\| \geq c\alpha \right] \leq P\left[\left\| Y\right\| \geq \alpha \right] \leq cP\left[\left\| X\right\| \geq c^{-1}\alpha \right] \label{eq:13} \end{equation} % for all $\alpha > 0$ , which proves (\ref{ineq:lem.main.4}). \section{The distribution of Banach valued Sidon series} To prove Theorem \ref{th:main}, Lemma \ref{lem:main} suggests that we consider independent copies of the given Sidon series. The construction of independent copies of a given trigonometric polynomial on a group $G$ is easily done on the product group. The spectra of the resulting polynomials are supported in a subset of the product of the character group. Our first goal in this section is to study properties of this set. To simplify the presentation, we will treat the commutative and noncommutative cases separately. Throughout this section $G$ will denote a compact abelian group with character group $\Gamma$ and Haar measure $\mu$. Similar meanings are attributed to $G_j , \Gamma_j , \mu_j$, respectively. \newtheorem{join}{Definition}[section] \begin{join} Suppose that $E_j\neq \emptyset$ is a subset of $\Gamma_j$, for $j = 1, \ldots , n$. The n-fold join of the sets $E_j$ is a subset of $\prod_{j=1}^{n}\Gamma_j$, denoted by $\bigvee\limits^n_{j=1}E_j$ , and defined by: $$\bigvee ^n_{j=1}E_j= \{ \gamma = (\gamma_1,\ldots , \gamma_n) \in \prod_{j=1}^n\Gamma_j : \ all\ but\ one\ \gamma_j \in E_j\ are\ 0\} .$$ If $E\neq\emptyset\subset\Gamma$, the n-fold join of $E$, denoted by $\bigvee_{j=1}^nE$, is the set $\bigvee_{j=1}^nE_j$, where $E_j=E$ for all $j=1, \ldots , n$. \label{def:join} \end{join} Thus a generic element $\gamma$ of $\bigvee_{j=1}^nE_j$ is of the form $\gamma = (0, 0, \ldots , \gamma_j , 0, \ldots , 0)$ where $\gamma_j \in E_j$. When $\gamma$ is evaluated at $x=(x_1, \ldots , x_j, \ldots , x_n) \in G^n$ we get: $$\gamma (x) = \gamma_j (x_j).$$ Suppose that $S\subset\Gamma$. For the sake of our proof of Theorem \ref{th:main}, it turns out that it is sufficient to study the $n$--fold join of the set $\{1\}\times S\subset \Z\times\Gamma$. What is needed is the following simple result. A more general result concerning joins of Sidon sets is presented following the proof of Theorem \ref{th:main}. \newtheorem{lem:sid1}[join]{Lemma} \begin{lem:sid1} Suppose $E\subset\Gamma$ is a Sidon set. Let $T=\{1\}\times E\subset\Z\times\Gamma$, and let $n$ be an arbitrary positive integer. The $n$-fold join $\bigvee_{j=1}^n T$ is a Sidon subset of $\Z^n\times\Gamma^n$ with Sidon constant $S\left(\bigvee_{j=1}^n T\right)$ equal to $S(E)$. \label{lem:sid1} \end{lem:sid1} \def\scj{$S\left(\bigvee_{j=1}^n T\right)$} \def\sidon{$\bigvee_{j=1}^n T$} {\bf Proof.} It is clear that $S(T)=S(E)$ and that \scj $\geq S(E)$. Let $F$ be a trigonometric polynomial with spectrum supported in \scj . We can write $F$ as: $$F =\sum_{j=1}^n f_j(t_j,x_j)$$ % where $$f_j=\sum_{l=1}^{k_j}a_{jl}e^{it_j}\chi_{jl}(x_j)= e^{it_j} \sum_{l=1}^{k_j}a_{jl}\chi_{jl}(x_j).$$ % For each $j=1,\ldots,n$, pick $x_j$ so that % $$\left|\sum_{l=1}^{k_j}a_{jl}\chi_{jl}(x_j)\right|=\|f_j\|_\infty,$$ % and then pick $t_j$ so that $$e^{it_j} \sum_{l=1}^{k_j}a_{jl}\chi_{jl}(x_j)=\|f_j\|_\infty .$$ % We have $$f_j(t_j,x_j) = \|f_j\|_\infty ,$$ and so $$F(t_1, t_2, \ldots, t_n, x_1, x_2, \ldots , x_n)= \sum_{j=1}^n\|f_j\|_\infty\geq\left(S(E)\right)^{-1} \sum_{j=1}^n\sum_{l=1}^{k_j}|a_{jl}|.$$ Hence (\ref{eq1}) holds for $F$\ with $c = S(E)$, and the proof is complete. \ We still need one ingredient for the proof of Theorem \ref{th:main}. This is the Khintchin--Kahane theorem. \newtheorem{th:KK}[join]{Theorem} \begin{th:KK}[{\cite[Th\'eor\eme K]{bib:Pi1}}] If $00$ that depends only on $S(E)$ such that for all $a=(a_n)_{n=1}^\infty\in\ell^2$ we have $$\mu\left[\left|\sum_{n=1}^\infty a_n\gamma_n\right| \geq cK_{1,2}(a,t)\right] \leq ce^{-c^{-1}t^2}$$ and $$\mu\left[\left|\sum_{n=1}^\infty a_n\gamma_n\right| \geq c^{-1}K_{1,2}(a,t)\right] \geq c^{-1}e^{-ct^2}$$ for all $t>0$. \end{th} Thus it is possible to calculate rearrangement invariant norms of scalar valued Sidon series in the spirit of Rodin and Semyonov \cite{bib:RS}. \def\tr{{\rm tr}\ } \ We note that these results generalize to noncommutative compact groups with no further difficulties. We follow the notation of \cite{bib:Pi2}, Section 5: $G$ is a compact group; $\Sigma$ is the dual object of $G$; $\mu$ is the normalized Haar measure on $G$. Thus $\Sigma$ is the set of equivalence classes of the irreducible representations of $G$. For each $\iota\in\Sigma$, we let $U_\iota$ denote a representing element of the equivalence class. Thus $U_\iota (x)$ is, for each $x\in G$, a unitary operator on a fixed finite dimensional Hilbert space $H_\iota$. The dimension of $H_\iota$ will be denoted by $d_\iota$. For further details, we refer the reader to \cite{bib:Pi2}, and \cite{bib:AHA2}. In analogy with Definition \ref{defsidon}, a subset $S\subset\Sigma$ is called a {\em Sidon set} if there is a constant $c$ such that $$\sum d_\iota\tr |\hat{f}(\iota)|\leq c\| f\|_\infty$$ for every $f\in{\cal C}_S(G)$, where ${\cal C}_S(G)$ is defined as in the abelian case. \def\eps{\varepsilon_\iota} Let $I$ be a countable indexing set. An analogue of the Rademacher functions is defined as a sequence, $\{\eps\}_{\iota\in I}$, of independent random variables, each $\eps$ being a random $d_\iota\times d_\iota$ orthogonal matrix, uniformly distributed on the orthogonal group ${\cal O}(d_\iota )$. These functions are studied in \cite{bib:Pi2} and \cite{bib:MP}. Note that we have the analogue of the Khintchin--Kahane inequality due to Pisier--Marcus, \cite[Corollary 2.12, p. 91]{bib:MP}. The analogue of Pisier's Theorem \ref{th:Pis1} can be easily established in this setting by repeating the proof in \cite{bib:Pi1} and making use of the properties of Sidon sets on noncommutative groups. All of these properties are found in \cite[Theorem (37.2)]{bib:AHA2}. For ease of reference, we state the result below, and omit the proof. \newtheorem{th:Pis1.noncom}[join]{Theorem} \begin{th:Pis1.noncom} Suppose that $P = (\iota) \subset \Sigma$ is a Sidon set, that $B$ is a Banach space, and that $M_\iota$ is a $d_\iota\times d_\iota$ matrix with values in $B$. Then there is a constant $c_1$, depending only on the Sidon constant $S(P)$, such that, for every $p \in [1, \infty [$, we have \begin{eqnarray} c_1^{-1}\left( {\bf E}\left\| \sum_{\iota\in F} d_\iota\tr (\eps M_\iota ) \right\|^p\right)^{\frac{1}{p}} & \leq & \left( \int_G\left\| \sum_{\iota\in F} d_\iota\tr (U_\iota M_\iota ) \right\|^p\right)^{\frac{1}{p}} \nonumber \\ & \leq & c_1\left( {\bf E}\left\| \sum_{\iota\in F} d_\iota\tr (\eps M_\iota )\right\|^p\right)^{\frac{1}{p}} \label{ineq:Pis.noncom} \end{eqnarray} for any finite subset $F$ of $P$ \label{th:Pis1.noncom} \end{th:Pis1.noncom} The result concerning the join of Sidon sets in duals of compact groups is a straight analogue of Theorem \ref{th:join}. We omit even the statement. We have thus all the necessary ingredients to prove a noncommutative version of our main Theorem \ref{th:main}. \newtheorem{th:main.noncom}[join]{Theorem} \begin{th:main.noncom} Let $G$ be a compact group with dual object $\Sigma$.\ Suppose that $P=\{\iota\}\subset\Sigma$ is a Sidon set. For $n = 1, \ldots, N$, let $a^n$ denote a $d_{\iota_n} \times d_{\iota_n}$ matrix with entries in a Banach space $B$. There is a constant $c>0$ such that, for all $\alpha >0$, we have: \begin{eqnarray} c^{-1}P\left[\left\|\sum_{n=1}^Nd_{\iota_n}\tr (\varepsilon_{\iota_n} a^n)\right\| \geq c\alpha\right] & \leq & \mu\left[\left\|\sum_{n=1}^Nd_{\iota_n}\tr (U_{\iota_n} a^n) \right\| \geq \alpha\right] \nonumber \\ & \leq & cP\left[\left\|\sum_{n=1}^Nd_{\iota_n}\tr (\varepsilon_{\iota_n} a^n)\right\| \geq c^{-1}\alpha\right]. \label{eq:main.noncom} \end{eqnarray} \label{th:main.noncom} \end{th:main.noncom} We close our paper by pointing out to the interested reader that, under suitable conditions, these methods also apply to commensurate sets of characters of Pelczy\'nski \cite{bib:Pe}, as indeed they do also to topological Sidon sets, also described by Pelczy\'nski {\em loc. sit.} \begin{thebibliography}{Dillo 83} \bibitem[HR]{bib:AHA2} E. Hewitt and K. A. Ross, Abstract Harmonic Analysis'', Vol. 2, {\bf 152}, Springer--Verlag, Berlin and New York, 1970. \bibitem[Ka]{bib:Ka} J.P. 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