Extensions of the Furstenberg-S\'ark\"ozy theorem via the arithmetic level-$d$ inequality

TL;DR

This paper extends Green and Sawhney's quasipolynomial bounds from square differences to general intersective polynomials, improving quantitative bounds for sets lacking such polynomial differences.

Contribution

It adapts the arithmetic level-$d$ inequality method to general intersective polynomials, allowing uniform effectiveness across iterative density increments.

Findings

Achieved quasipolynomial upper bounds for sets avoiding polynomial differences.

Extended the method to handle changing polynomials at each iteration.

Developed weighted exponential sum estimates of Rice.

Abstract

Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--S\'ark\"ozy theorem for square differences by proving an ''arithmetic level-$d$'' inequality, thereby yielding a greatly improved density increment scheme. We adapt their method to general intersective polynomials $h\in\mathbb{Z}[x]$ and obtain an analogous quasipolynomial upper bound for the largest subset of $\{1,2,\dots,X\}$ whose difference set contains no nonzero element of the form $h(n)$ with $n\in \mathbb{Z}$. This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences. In contrast to the square case, extending the method to general intersective polynomials requires performing a density increment iteration in which the underlying polynomial changes at each step; a key contribution of this paper is to show that the arithmetic level-$d$ inequality remains effective uniformly across all auxiliary polynomials arising in the iteration. We also develop smoothly weighted versions of the exponential sum estimates of Rice.