Packing index of subsets in Polish groups

TL;DR

This paper investigates the packing index of subsets in Polish groups, revealing that for certain subsets the packing index is constrained to specific cardinal values, and constructs examples illustrating these properties.

Contribution

It establishes bounds on the packing index for various subsets in Polish groups and constructs examples with prescribed packing indices, advancing understanding of geometric smallness in these groups.

Findings

Packing index of sigma-compact subsets is in
or countable or continuum.

Borel subsets cannot have packing indices in a certain uncountable interval.

Existence of nowhere dense Haar null sets with arbitrary large packing index.

Abstract

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 $\kappa\in[4,\cc]$.