On random perturbation of some intersective sets.
Let \(S\) be a subset of \(\mathbb{Z}\) and let \(I= (I_n)_{n\geq 1}\) be a sequence of intervals in \(\mathbb{Z}\) whose lengths are increasing. Set \[ b(S, I)= \limsup_{n\to\infty}\, {|S\cap I_n|\over|I_n|}. \] The Banach density of \(S\), denoted by \(b(S)\), is defined by \(b(S)= \sup_Ib(S,I)\), where the supremum is taken over all sequences \(I\) of intervals in \(\mathbb{Z}\) whose lengths increase. A sequence of nonnegative integers \({\mathbf k}= (k_n)_{n\geq 1}\) is said to be multiply intersective if given any subset \(E\) of \(\mathbb{N}\) with positive Banach density \(b(E)\), there exists a subset \(R\) of \(\mathbb{N}\) with \[ \lim_{N\to\infty}\, {|R\cap[1, N]|\over N}\geq b(E), \] such that for any subset \(\{n_1,\dots, n_r\}\) of \(R\) we have \[ b(E\cap (E+ k_{n_1})\cap\cdots\cap(E+ k_{n_r}))> 0. \] The authos prove the following result: If \({\mathbf k}= (k_n)_{n\geq 1}\) is a multiply intersective set, then this is also the case of \((k_n+ \theta_n)_{n\geq 1}\), for almost all \(\mathbb{N}\)-valued independent identically distributed random variables \(\theta= (\theta_n)_{n\geq 1}\).
scientific article; zbMATH DE number 2124777
scientific article