On density of infinite subsets I

TL;DR

This paper investigates the density properties of infinite subsets under group actions on compact metric spaces, demonstrating that certain transformations can make these subsets arbitrarily dense, with applications to tori and interval exchange transformations.

Contribution

It establishes new results on the density of infinite subsets under group actions, including the existence of transformations making subsets dense and generic behavior in interval exchange transformations.

Findings

Existence of transformations making subsets epsilon-dense in tori.

Generic 3-IETs lead to dense unions of iterates of subsets.

For any infinite subset of [0,1], the Hausdorff distance to the whole interval can be made arbitrarily small.

Abstract

Let $Y$ be a compact metric space, $G$ be a group acting by transformations on $Y$. For any infinite subset $A\subset Y$, we study the density of $gA$ for $g\in G$ and quantitative density of the set $\displaystyle{\bigcup_{g\in G_n}gA}$ by the Hausdorff semimetric $d^H$. It is proven that for any integer $n\ge 2$, $\epsilon>0$, any infinite subset $A\subset \mathbb T^n$, there is a $g\in SL(n,\mathbb Z)$ such that $gA$ is $\epsilon$-dense. We also show that, for any infinite subset $A\subset [0,1]$, for generic rotation and generic 3-IET, $$\liminf_nn\cdot d^H\left(\bigcup_{k=0}^{n-1}T^kA,[0,1]\right)=0.$$