Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality

Abstract

Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--Sárközy 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.