Entiers al\'eatoires, ensembles de Sidon, densit\'e dans le groupe de Bohr et ensembles d'analyticit\'e

TL;DR

This paper investigates the properties of randomly selected integer sequences, showing that their density and algebraic properties depend on the asymptotic behavior of selection probabilities, with implications for Sidon sets and sets of analyticity.

Contribution

It characterizes the almost sure properties of random integer sequences in relation to Sidon sets and sets of analyticity based on their selection probabilities.

Findings

If n*ω_n is bounded, the sequence is almost surely a Sidon set and non-dense in the Bohr group.

If n*ω_n tends to infinity, the sequence is almost surely a set of analyticity and dense in the Bohr group.

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 $\limsup_{n\to\infty}n\varpi_{n}<\infty$, $\Lambda$ 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.