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Quasirandom group actions

Let $G$ be a finite group acting transitively on a set $Ω$. 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 $Ω$. This convolution bound allows us to give sufficient conditions such that sets $S,T\subset G$ and $Γ\subseteq Ω$ contain elements $s\in S, t\in T, γ\inΓ$ such that $s(γ)=t$. 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.