We show that if $h\in\mathbb{ Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{ q}$ has a solution for every positive integer $q$, then any subset of $\{ 1,2,\ldots,N\}$ 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.
Categories: math.NT