Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On density of infinite subsets I   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, $$\liminf_nn\cdot d^H\left(\bigcup_{k=0}^{n-1}T^kA,[0,1]\right)=0.$$

12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
18
The categories of this paper are: econ