Bohr recurrence and density of non-lacunary semigroups of ℕ

Abstract

A subset R R of integers is a set of Bohr recurrence if every rotation on T d \mathbb {T}^d returns arbitrarily close to zero under some non-zero multiple of R R . We show that the set { k ! 2 m 3 n : k , m , n ∈ N } \{k!\, 2^m3^n\colon 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 P is a real polynomial with at least one non-constant irrational coefficient, then the set { P ( 2 m 3 n ) : m , n ∈ N } \{P(2^m3^n)\colon m,n\in \mathbb {N}\} is dense in T \mathbb {T} , thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.