Good Bounds for Sets Lacking Skew Corners

Abstract

A skew corner is a triple of points in Z×Z of the form (x,y),(x,y+a) and (x+a,y′). Pratt posed the following question: how large can a set A⊆[n]×[n] be, provided it contains no non-trivial skew corner (i.e. one for which a≠0)? We prove that |A|≤exp(-clogcn)n2, for an absolute constant c>0, which, along with a construction of Beker, essentially resolves Pratt’s question. Our argument 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.