QUASIRANDOM GROUP ACTIONS

Nick Gill

doi:10.1017/fms.2016.8

Abstract

Let $G$ be a finite group acting transitively on a set $\unicode[STIX]{x1D6FA}$. We study what it means for this action to be 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 $\unicode[STIX]{x1D6FA}$. This convolution bound allows us to give sufficient conditions such that sets $S\subseteq G$ and $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2}\subseteq \unicode[STIX]{x1D6FA}$ contain elements $s\in S,\unicode[STIX]{x1D714}_{1}\in \unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D714}_{2}\in \unicode[STIX]{x1D6E5}_{2}$ such that $s(\unicode[STIX]{x1D714}_{1})=\unicode[STIX]{x1D714}_{2}$. 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.

Highlights

In his seminal 2008 paper entitled ‘Quasirandom groups’, Gowers introduced the notion of a d-quasirandom group
We study a particular instance of such an action in order to prove the following sum-product result for large sets in finite fields
Our work suggests that these ideas belong more properly in the more general setting of d-quasirandom group actions

Summary

Main results

In order to state our main results we must establish some notation which will hold throughout the paper. We set G to be a finite group acting transitively on a finite set Ω. Consider two functions X : G → R and Y : Ω → R. We define the convolution X ∗c Y of X and Y to be the following function on Ω:. This definition, which has appeared in various places in the literature, is a generalization of the definition of convolution given in [2]. We write H = StabG(ω), the stabilizer in G of some element ω ∈ Ω. If χ is a representation of H , The representation 1GH is we the write χ

G H for permutation the representation of representation of G

Large sets grow