Quasirandom group actions

TL;DR

This paper introduces the concept of quasirandomness in finite group actions, connecting it to convolution bounds and exploring implications for combinatorial properties, expanders, and finite simple groups.

Contribution

It generalizes Gowers' quasirandomness to group actions, providing new bounds and applications in combinatorics and group theory.

Findings

Established convolution bounds for quasirandom group actions

Derived conditions for element existence in large subsets

Extended Gowers' trick and proved sum-product type results

Abstract

Let $G$ be a finite group acting transitively on a set $\Omega$. We study what it means for this action to be {\it quasirandom}, thereby generalizing Gowers' study of quasirandomness in groups. We connect this notion of quasirandomness to an upper bound for the convolution of functions associated with the action of $G$ on $\Omega$. This convolution bound allows us to give sufficient conditions such that sets $S,T\subset G$ and $\Gamma\subseteq \Omega$ contain elements $s\in S, t\in T, \gamma\in\Gamma$ such that $s(\gamma)=t$. Other consequences include an analogue of `the Gowers trick' of Nikolov and Pyber for general group actions, a sum-product type theorem for large subsets of a finite field, as well as applications to expanders and to the study of the diameter and width of a finite simple group.