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 $\{k!2^m{3}^n\text{\hspace{0.17em}⁣}:k,m,n\in \mathbb{N}\}$ 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 $\{P(2^m{3}^n)\text{\hspace{0.17em}⁣}:m,n\in \mathbb{N}\}$ is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.