Weakly mixing sets and polynomial equations

We investigate polynomial patterns which can be guaranteed to appear in \emph{weakly mixing} sets introduced by introduced by Furstenberg and studied by Fish. In particular, we prove that if $A\subset \mathbb{N}$ is a weakly mixing set and $p(x)\in \mathbb{Z}[x]$ a polynomial of odd degree with positive leading coefficient, then all sufficiently large integers $N$ can be represented as $N=n_1+n_2$, where $p(n_1)+m,p(n_2)+m\in A$ for some $m\in A$.