  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.
