Entiers aléatoires, ensembles de Sidon, densité dans le groupe de Bohr et ensembles d'analyticité

Abstract

We study properties of a sequence $\Lambda$ obtained by a randomselection of integers $n$, where $n\in \Lambda$ with probability ${\varpi }_n$, independently of the other choices. We distinguish two cases : if ${\mathrm{limsup}}_{n\to \infty }n{\varpi }_n<\infty$, $\Lambda$ is a.s. a Sidon set, non-dense in the Bohr group ; if ${\lim }_{n\to \infty }n{\varpi }_n=\infty$, then $\Lambda$ is a.s. a set of analyticity and is dense in the Bohr group.