## 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.)

### Keywords:

compact abelian groups; Sidon set; quasi-independent sets

Zbl 0351.43008
Full Text:

### References:

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