Glasner property for linear group actions and their products

Abstract

AbstractA theorem of Glasner from 1979 shows that if $$Y \subset \mathbb {T}= \mathbb {R}/\mathbb {Z}$$ Y ⊂ T = R / Z is infinite then for each $$\epsilon > 0$$ ϵ > 0 there exists an integer n such that nY is $$\epsilon $$ ϵ -dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on $$\mathbb {T}^d$$ T d also satisfy such a Glasner property where each infinite set (in fact, sufficiently large finite set) will have an $$\epsilon $$ ϵ -dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski connected Zariski closure. This makes use of recent results on linear random walks on the torus. We also pose a natural question that asks whether the Cartesian product of two actions satisfying the Glasner property also satisfy a Glasner property for infinite subsets which contain no two points on a common vertical or horizontal line. We answer this question affirmatively for many such Glasner actions by providing a new Glasner-type theorem for linear actions that are not irreducible, as well as polynomial versions of such results.