New bounds for the Furstenberg-S\'ark\"ozy theorem

TL;DR

This paper establishes new upper bounds on the size of subsets of integers that contain no two elements differing by a perfect square, improving understanding of additive combinatorics related to polynomial differences.

Contribution

It provides novel bounds for sets avoiding square differences, advancing the theoretical limits in the Furstenberg-Sárközy theorem.

Findings

Sets with no two elements differing by a perfect square are significantly smaller than the total set size.

The bound on the size of such sets is exponentially decreasing in the square root of the logarithm of N.

The results refine previous bounds and contribute to the understanding of polynomial difference problems.

Abstract

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by a square. Then $|A| \ll N e^{-c\sqrt{\log N}}$.