Ajtai-Szemer\'edi Theorems over quasirandom groups

TL;DR

This paper extends the Ajtai-Szemerédi Theorem to quasirandom non-Abelian groups, demonstrating that dense subsets contain many specific patterns and that the set of differences with these patterns is syndetic.

Contribution

It establishes strong pattern existence results in Cartesian squares of quasirandom groups, including syndeticity of good difference sets, for the first time in this context.

Findings

Dense subsets contain many patterns for most differences

The set of good differences is syndetic in certain cases

Results apply to non-Abelian quasirandom groups

Abstract

Two versions of the Ajtai-Szemer\'edi Theorem are considered in the Cartesian square of a finite non-Abelian group $G$. In case $G$ is sufficiently quasirandom, we obtain strong forms of both versions: if $E \subseteq G\times G$ is fairly dense, then $E$ contains a large number of the desired patterns for most individual choices of `common difference'. For one of the versions, we also show that this set of good common differences is syndetic.