Packing Index of Subsets in Polish Groups

Abstract

For a subset A of a Polish group G, we study the (almost) packing index pack( A) (respectively, Pack( A)) of A, equal to the supremum of cardinalities |S| of subsets $S\subset G$ such that the family of shifts ${\left\{xA\right\}}_{x\in S}$ is (almost) disjoint (in the sense that for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are large in a geometric sense. We show that $\mathrm{pack}\left(A\right)\in \mathbb{N}\cup \left\{{\aleph }_0,\mathfrak{c}\right\}$ for any σ-compact subset A of a Polish group. In each nondiscrete Polish Abelian group G we construct two closed subsets $A,B\subset G$ with $\mathrm{pack}\left(A\right)=\mathrm{pack}\left(B\right)=\mathfrak{c}$ and $\text{Pack}(A\cup B)=1$ and then apply this result to show that G contains a nowhere dense Haar null subset $C\subset G$ with pack(C)=Pack(C)=κ for any given cardinal number $\kappa \in \left[4,\mathfrak{c}\right]$.