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

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^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 is a real polynomial with at least one non-constant irrational coefficient, then the set \{P(2^m3^n)\colon 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. [ABSTRACT FROM AUTHOR]