ε-dense images of large sets under a group action

Wednesday, 26 April 2023

Abstract

A theorem of Glasner states that if Y is an infinite subset of the torus (real numbers mod 1) then for each ε>0 there exists an integer n such that nY is ε-dense. One can ask whether this holds for other semigroup actions: We say an action of a semigroup G on a compact metric space X is N(ε)-Glasner if each ε>0 and subsets Y of X with cardinality |Y|>N(ε) there exists g in G such that gY is ε-dense in X. We show how the equidistribution of random walks can be used to demonstrate that many linear group actions on the torus are 1/εᶜ Glasner (for some c>0). Based on joint work with Sasha Fish.