A maximal extension of the Bloom-Maynard bound for sets with no square differences

We show that if $h\in \mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}q)$ has a solution for every positive integer $q$, then any subset of $\{1,2,\dots ,N\}$ with no two distinct elements with difference of the form $h(n)$, with $n$ positive integer, has density at most $(\log N{)}^{-c\log \log \log N}$, for some constant $c$ that depends only on $k$. This improves on the best bound in the literature, due to Rice, and generalizes a recent result of Bloom and Maynard.