Sidon sets and Riesz products
type: Item
instance of: scholarly article
title: Sidon sets and Riesz products
zbMATH Open document ID: 0578.43008
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74662
review text: A new arithmetical characterization of Sidon sets in discrete abelian groups is given: (*) \(\Lambda \subset \Gamma\) is a Sidon set iff there is \(\delta >0\) such that if \((a_{\gamma})_{\gamma \in \Lambda}\) is a finite sequence of scalars, there exists a quasi-independent subset A of \(\bigwedge\) such that \(\sum_{\gamma \in A}| a_{\gamma}| \geq \delta \sum_{\gamma \in \Lambda}| a_{\gamma}|\). (A set is called quasi-independent if \(\sum_{A}z_{\gamma} \gamma =0\) \((z_{\gamma}=-1,0,1)\) implies \(z_{\gamma}=0\) for each \(\gamma \in A.)\) Condition (*) is formally stronger than Pisier's condition (**) which is obtained from (*) by letting \(a_{\gamma}=1\) for each \(\gamma\) [\textit{G. Pisier}, Adv. Math., Suppl. Stud. 7B, 685-726 (1981; Zbl 0468.43008)]. The author first reproves (**) for Sidon sets deducing it by means of a probabilistic argument from another characterization of them, namely \(\| \sum a_{\gamma} \gamma \|_ p\leq C p^{1/2}(\sum | \alpha_{\gamma}|^ 2)^{1/2}\), \(p\geq 1\) (''only if'' by Rudin, ''if'' by Pisier). An analogous method serves for proving the following key lemma: if \(\bigwedge_ 1,...,\bigwedge_ j\) are disjoint quasi- independent sets in \(\Gamma\) and \(| \bigwedge_{j+1}| /| \bigwedge_ j| >R\) \((j=1,...,J-1)\) with sufficiently large constant R then there are subsets \(\bigwedge'_ j\) of \(\bigwedge_ j\) such that \(| \bigwedge'_ j| >1/10| \bigwedge_ j|\) and \(\cup^{J}_{j=1}\bigwedge'_ j\) is quasi-independent. This allows to deduce (*) relatively easily from (**) with \(\delta\) of the same order of magnitude. As a corollary from this result the author obtains the following interpolation theorem: if \(\bigwedge\) is a Sidon set, there is a \(\delta >0\) such that whenever \((a_{\gamma})_{\gamma \in \bigwedge}\) is a finite scalar sequence and \(| a_{\gamma}| \leq \delta\) then there is a \(\mu\) in the \(\sigma\)-convex hull of a sequence of Riesz products such that \({\hat\mu}(\gamma) = a_{\gamma}\) (\(\gamma\in \Lambda)\). …
reviewed by: Stanisław Hartman
Mathematics Subject Classification ID: 43A46
zbMATH DE Number: 3925511
zbMATH Keywords: Sidon sets
zbMATH Keywords: quasi-independent sets
zbMATH Keywords: Riesz products
author: Jean Bourgain
MaRDI profile type: Publication
OpenAlex ID: W1967245832
cites work: Propriétés de décomposition pour les ensembles de Sidon
cites work: Ensembles de Sidon topologiques
cites work: Q4121006
cites work: Q4157128
cites work: Q3921655
cites work: Q3326227
cites work: Q3265794
DOI: 10.5802/AIF.1003
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 3925511