Corners in Quasirandom Groups via Sparse Mixing

TL;DR

This paper advances bounds on corner-free sets in quasirandom groups and improves lower bounds on communication complexity in a multi-party model, based on a new combinatorial theorem extending recent progress in additive combinatorics.

Contribution

It introduces a general combinatorial theorem that significantly improves bounds on corner-free sets and communication complexity, building on recent breakthroughs in additive combinatorics.

Findings

Upper bounds on corner-free sets improved to quasi-polynomial

Lower bounds on communication complexity substantially increased

Theoretical framework extends recent combinatorial results

Abstract

We improve the best known upper bounds on the density of corner-free sets over quasirandom groups from inverse poly-logarithmic to quasi-polynomial. We make similarly substantial improvements to the best known lower bounds on the communication complexity of a large class of permutation functions in the 3-player Number-on-Forehead model. Underpinning both results is a general combinatorial theorem that extends the recent work of Kelley, Lovett, and Meka (STOC'24), itself a development of ideas from the breakthrough result of Kelley and Meka on three-term arithmetic progressions (FOCS'23).