Quasirandom group actions

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.