  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 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 \textit{Glasner 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.
