Bohr recurrence and density of non-lacunary semigroups of $\mathbb{N}$

Abstract

A subset $R$ of integers is a set of Bohr recurrence if every rotation on ${\mathbb{T}}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!2^m{3}^n:k,m,n\in \mathbb{N}\}$ is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if $P$ is a real polynomial with at least one non-constant irrational coefficient, then the set $\{P(2^m{3}^n):m,n\in \mathbb{N}\}$ is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.

13 pages. Added a comment after Theorem 3 and a reference. To appear in the Proceedings of the AMS