Good bounds for sets lacking skew corners

Abstract

A skew corner is a triple of points in $\mathbb{Z}\times \mathbb{Z}$ of the form $(x,y),(x,y+a)$ and $(x+a,y^{\prime })$. Pratt posed the following question: how large can a set $A\subseteq [n]\times [n]$ be, provided it contains no non-trivial skew corner (i.e. one for which $a\ne 0$)? We prove that $|A|\le \exp (-c{\log }^{c}n)n^2$, for an absolute constant $c>0$, which, along with a construction of Beker, essentially resolves Pratt's question. Our argument is represents a two-dimensional variant of the method of Kelley and Meka, which they used to prove Behrend-type bounds in Roth's theorem. A very similar result was obtained independently and simultaneously by Jaber, Lovett and Ostuni.

28 pages, added the reference to a similar result of Jaber, Lovett and Ostuni, which was obtained simultaneously