English

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

Number Theory 2025-01-13 v2

Abstract

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

Keywords

Cite

@article{arxiv.2411.17448,
  title  = {New bounds for the Furstenberg-S\'ark\"ozy theorem},
  author = {Ben Green and Mehtaab Sawhney},
  journal= {arXiv preprint arXiv:2411.17448},
  year   = {2025}
}

Comments

35 pages; new version improves exponent from $1/4 - o(1)$ to $1/2$