Glasner property for linear group actions and their products

A theorem of Glasner from 1979 shows that if $Y\subset \mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ is infinite then for each $\epsilon >0$ there exists an integer $n$ such that $nY$ is $\epsilon$-dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on ${\mathbb{T}}^d$ also satisfy such a $\mathit{\text{Glasner}}\mathit{\text{property}}$ where each infinite set (in fact, arbitrarily large finite set) will have an $\epsilon$-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.