A new bound for the Furstenberg--Sárközy theorem using the van der Corput property

Source abstract

We show that if $A\subseteq \mathbb{N}\cap[1,N]$ has no nonzero square difference, then $$|A|\ll N\exp(-c\sqrt{\log N\log\log N}),$$ improving upon a recent result of Green and Sawhney. The proof exploits a quantitative version of the van der Corput property with signed coefficients and builds on previous constructions of Slijepčević, Slijepčević--Ninčević, and Fan-Lott. The proof of the upper bound is elementary and self-contained. We also prove a matching lower bound for the constant coefficient of any van der Corput witness for squares, showing that our quantitative van der Corput bound is sharp up to the constant $c$.