The Furstenberg-S\'ark\"ozy theorem for polynomials in one or more prime variables

TL;DR

This paper extends the Furstenberg-Sárközy theorem to polynomial differences involving prime variables, providing upper bounds on large subsets of integers lacking such prime-based polynomial differences.

Contribution

It establishes upper bounds for subsets avoiding prime polynomial differences, generalizing previous results to multiple prime variables.

Findings

Derived bounds match known results for unrestricted integers.

Extended Furstenberg-Sárközy theorem to prime-variable polynomials.

Unified approach for multiple 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 $\ell \geq 2$.