Triangles in cartesian squares of quasirandom groups

TL;DR

This paper demonstrates that large subsets of cartesian squares of finite quasirandom groups, including finite simple groups, contain many triangular configurations, supported by a new double recurrence theorem for certain measure-preserving group actions.

Contribution

It introduces a novel double recurrence theorem for non-amenable, minimally almost periodic groups, establishing the abundance of triangles in quasirandom group structures.

Findings

Triangular configurations are abundant in large subsets of cartesian squares.

The results apply to classes of groups with the quasirandom ultraproduct property.

A new recurrence theorem for non-amenable groups is established.

Abstract

We prove that triangular configurations are plentiful in large subsets of cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.