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A theorem of Glasner from 1979 shows that if $Y \subset \mathbb{T} =\mathbb{R}/\mathbb{Z}$ is infinite then for each $\epsilon > 0$ there exists aninteger $n$ such that $nY$ is $\epsilon$-dense. This has been extended invarious works by showing that certain irreducible linear semigroup actions on$\mathbb{T}^d$ also satisfy such a \textit{Glasner property} where eachinfinite set (in fact, arbitrarily large finite set) will have an$\epsilon$-dense image under some element from the acting semigroup. We improvethese works by proving a quantitative Glasner theorem for irreducible lineargroup actions with Zariski-connected Zariski-closure. This makes use of recentresults on linear random walks on the torus. We also pose a natural questionthat asks whether the cartesian product of two actions satisfying the Glasnerproperty also satisfy a Glasner property for infinite subsets which contain notwo points on a common vertical or horizontal line. We answer this questionaffirmatively for many such Glasner actions by providing a new Glasner-typetheorem for linear actions that are not irreducible, as well as polynomialversions of such results.

Glasner property for linear group actions and their products