Talk information

Date: Sunday, June 2, 2024
Time: 10:10–11:00
Place: Schreiber 309
Speaker: Shachar Lovett (UCSD)
Title: Skew corner free sets

Abstract:

The corners problem of Ajtai and Szemerédi is a classical problem in additive combinatorics: what is the largest density of a set in the 2-$D$ grid $[N]\times[N]$ that does not contain a corner? Here, a corner is a triple of points of the form $(x,y),(x,y+h),(x+h,y)$ where $h$ is nonzero. Despite decades of research, the gap between the best upper and lower bounds is double-exponential. Recently, Pratt introduced a variant of the problem, where corners are replaced by “skew corners”, which are triples of points of the form $(x,y),(x,y+h),(x+h,y’)$. His motivation was an intriguing connection to matrix multiplication algorithms. Following a sequence of works this year, it turns out that the answer here is much better understood than in the classical corners problem, with nearly matching upper and lower bounds. I will describe these works and some of the proof ideas that go into them. In particular, I will discuss a joint work with Michael Jaber and Anthony Ostuni where we prove the upper bound.