Multiple recurrence in quasirandom groups

TL;DR

This paper proves a new mixing theorem for quasirandom groups, showing that certain quadruples behave like random tuples, using ultraproducts and ultrafilter techniques to handle the non-elementary proof.

Contribution

It introduces an ultraproduct framework and ultrafilter methods to establish mixing properties in quasirandom groups, advancing understanding of their combinatorial structure.

Findings

Established a mixing theorem for quasirandom groups

Developed an ultraproduct approach for infinitary analysis

Presented recurrence theorems involving group tuples

Abstract

We establish a new mixing theorem for quasirandom groups (finite groups with no low-dimensional unitary representations) $G$ which, informally speaking, asserts that if $g, x$ are drawn uniformly at random from $G$, then the quadruple $(g,x,gx,xg)$ behaves like a random tuple in $G^4$, subject to the obvious constraint that $gx$ and $xg$ are conjugate to each other. The proof is non-elementary, proceeding by first using an ultraproduct construction to replace the finitary claim on quasirandom groups with an infinitary analogue concerning a limiting group object that we call an \emph{ultra quasirandom group}, and then using the machinery of idempotent ultrafilters to establish the required mixing property for such groups. Some simpler recurrence theorems (involving tuples such as $(x,gx,xg)$) are also presented, as well as some further discussion of specific examples of ultra quasirandom groups.