Sidon sets and Riesz products
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 \({\hat\mu}(\gamma) = a_{\gamma}\) (\(\gamma\in \Lambda)\). The proof is reduced to a hint: a duality argument shall be used. Another corollary states that a Sidon set \(\bigwedge\) tending to infinity (it means that given a finite set \(\Gamma_ 0\) one has \(\gamma -\delta \not\in \Gamma_ 0\) if \(\gamma\) \(\neq \delta\) and \(\gamma\),\(\delta\in \bigwedge\) are beyond a finite set) is of first type, in other words there exists a constant C such that for each open subset I of \(G={\hat \Gamma}\) there is a finite subset \(\bigwedge_ 0\) of \(\bigwedge\) such that \(\sum_{\gamma \in \bigwedge \setminus \bigwedge_ 0}| \alpha_{\gamma}| \leq C \| \sum_{\gamma \in \bigwedge \setminus \bigwedge_ 0}\alpha_{\gamma} \) \(\gamma \|_{C(I)}\) for finite scalar sequences \((\alpha_{\gamma})\). In the proof an argument is used that can also be adapted (actually simplified) for deducing sidonicity directly from (*) without passing through (**). Condensed proofs, some disorder in notation and numerous misprints make the paper rather difficult to read.
scientific article; zbMATH DE number 3925511
scientific article