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 $σ$-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 $[\sq(Π^1_1),\cc)$ where $\sq(Π^1_1)$ 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 $\ind_P(C)=\Ind_P(C)=κ$ for any given cardinal number $κ\in[4,\cc]$.

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