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We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such thatthe 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 withdifference 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 generalizesa recent result of Bloom and Maynard.

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