On density of infinite subsets II: dynamics on homogeneous spaces

TL;DR

This paper investigates the density properties of infinite subsets under group actions on homogeneous spaces and tori, demonstrating that such sets can be made arbitrarily dense through group transformations.

Contribution

It establishes that for any infinite subset in certain homogeneous spaces, a group element exists to make the set arbitrarily dense, extending to specific discrete actions on tori.

Findings

Any infinite subset can be transformed to be epsilon-dense in the space.

The result applies to actions of minimal parabolic subgroups on homogeneous spaces.

Similar density results are shown for certain discrete group actions on tori.

Abstract

Let $G$ be a noncompact semisimple Lie group, $\Gamma$ be an irreducible cocompact lattice in $G$, and $P<G$ be a minimal parabolic subgroup. We consider the dynamics of $P$ acting on $G/\Gamma$ by left translation. For any infinite subset $A\subset G/\Gamma$, we show that, for any $\epsilon>0$, there is a $g\in P$ such that $gA$ is $\epsilon$-dense. We also prove a similar result for certain discrete group actions on $\mathbb T^n$.