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A maximal extension of the best-known bounds for the Furstenberg–Sárközy theorem

A maximal extension of the best-known bounds for the Furstenberg–Sárközy theorem

We show that if $h\in \mathbb Z[x]$ is a polynomial of degree $k\geq 2$ such that $h(\mathbb N)$ contains a multiple of $q$ for every $q\in\mathbb N$, known as an <em>intersective polynomial</em>, then any subset of $\{1,\dots,N\}$ with no nonzero differe