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Propriétés de décomposition pour les ensembles de Sidon. (French) Zbl 0546.43006

The paper consists of two parts. The main result of the first is the following: if \(G_ 1\) and \(G_ 2\) are compact abelian groups and \(\Lambda\) a Sidon set in \((G_ 1\times G_ 2)^{\wedge}\) then \(\Lambda\) is a finite union of sets \(\Lambda_{\alpha}\) such that, for each \(\alpha\), the projections of \(\Lambda_{\alpha}\) on \(\hat G_ 1\) and on \(\hat G_ 2\) are injective and both are Sidon sets in \(\hat G_ 1\) and \(\hat G_ 2\) respectively. As a corollary the author derives the following extension of the known theorem of Malliavin-Brameret and Malliavin: for any square-free n, every Sidon set \(\Lambda\) in the direct sum \(\sum {\mathcal Z}(n)\) is a finite union of quasi-independent sets \(\Lambda_{\alpha}\), this means that there is no relation of the form \(\sum_{\gamma_ j\in\Lambda_{\alpha}\backslash\{0\}}\epsilon_ j\gamma_ j=0 (\epsilon_ j=\pm 1)\). The method of proof is based on the probabilistic characterization of Sidon sets by Rider and on some related results. In the second part the author calls a set \(A\subset\hat G \ell\)-quasi-independent if any relation \(\sum_{\gamma\in A}\epsilon_{\gamma}\gamma =0\) with \(\epsilon_{\gamma}\in\{-1,0,1\}\) and \(\sum| \epsilon_{\gamma}|\leq \ell\) is trivial and he proves that, for each \(\ell\), every Sidon set is a finite union of \(\ell\)-quasi- independent sets. He begins by proving a combinatorial theorem, interesting in itself, which reads: let \(S_{\ell}(A)=\{\sum_{\gamma\in A}\epsilon_{\gamma}\gamma:\quad\sum |\epsilon_{\gamma}|\leq \ell\};\) there exists a function \(\Phi\) (\(\ell,C)\) such that if \(\Lambda \subset\hat G\) and \(| S_{\ell}(A)\cap\Lambda |\leq C| A|\) for any finite set \(A\subset\Lambda \), then \(\Lambda\) is a finite union of \(\Phi\) (\(\ell,C) (\ell +1)\)-quasi-independent sets. Since this theorem applies in particular to Sidon sets, the above-mentioned result follows. As a corollary one obtains the statement that every Sidon set is a finite union of sets tending to infinity (for the definition see the book of J. M. López and K. A. Ross [Sidon sets (1975; Zbl 0351.43008)], where this statement is proved for countable Sidon sets and left as an open question for the other ones).
Reviewer: S.Hartman

MSC:

43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

Citations:

Zbl 0351.43008
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References:

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