  We prove that a sumset of a TE subset of (\N) (these sets can be viewed as
"aperiodic" sets) with a set of positive upper density intersects a set of
values of any polynomial with integer coefficients., i.e. for any (A \subset \N
) a TE set, for any (p(n) \in \Z[n]: \deg{p(n)} > 0, p(n) \to_{n \to \infty}
\infty ) and any subset (B \subset \N ) of positive upper density we have (R_p
= A+B \cap \{p(n) | n \in \N \} \neq \emptyset). For (A ) a WM set (subclass of
TE sets) we prove that (R_p ) has lower density 1. In addition we obtain a
generalization of the latter result to the case of several polynomials and
several WM sets.
