Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Packing index of subsets in Polish groups   For a subset $A$ of a Polish group $G$, we study the (almost) packing index
$\ind_P(A)$ (resp. $\Ind_P(A)$) of $A$, equal to the supremum of cardinalities
$|S|$ of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$
is (almost) disjoint (in the sense that $|xA\cap yA|<|A|$ for any distinct
points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are
small in a geometric sense. We show that $\ind_P(A)\in \IN\cup\{\aleph_0,\cc\}$
for any $\sigma$-compact subset $A$ of a Polish group. If $A\subset G$ is
Borel, then the packing indices $\ind_P(A)$ and $\Ind_P(A)$ cannot take values
in the half-interval $[\sq(\Pi^1_1),\cc)$ where $\sq(\Pi^1_1)$ is a certain
uncountable cardinal that is smaller than $\cc$ in some models of ZFC. In each
non-discrete Polish Abelian group $G$ we construct two closed subsets
$A,B\subset G$ with $\ind_P(A)=\ind_P(B)=\cc$ and $\Ind_P(A\cup B)=1$ and then
apply this result to show that $G$ contains a nowhere dense Haar null subset
$C\subset G$ with $\ind_P(C)=\Ind_P(C)=\kappa$ for any given cardinal number
$\kappa\in[4,\cc]$.

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The categories of this paper are: eess
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
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The categories of this paper are: cmp-lg