The Furstenberg-Sárközy theorem for polynomials in one or more prime variables

Abstract

We establish upper bounds on the size of the largest subset of $\{1,2,\dots ,N\}$ lacking nonzero differences of the form $h(p_1,\dots ,p_{\ell })$, where $h\in \mathbb{Z}[x_1,\dots ,x_{\ell }]$ is a fixed polynomial satisfying appropriate conditions and $p_1,\dots ,p_{\ell }$ are prime. The bounds are of the same type as the best-known analogs for unrestricted integer inputs, due to Bloom-Maynard and Arala for $\ell =1$, and to the authors for $\ell \ge 2$.

22 pages. arXiv admin note: text overlap with arXiv:2006.15400