Packing Index of Subsets in Polish Groups

For a subset A of a Polish group G, we study the (almost) packing index indP (A) (resp. IndP (A)) of A, equal to the supremum of cardinalities |S| of subsets S ⊂ G such that the family of shifts {xA}x∈S is (almost) disjoint (in the sense that |xA ∩ yA| < |A| for any distinct points x, y ∈ S). Subsets A ⊂ G with small (almost) packing index are small in a geometric sense. We show that indP (A) ∈ N ∪ {א0, c} for any σ-compact subset A of a Polish group. If A ⊂ G is Borel, then the packing indices indP (A) and IndP (A) cannot take values in the half-interval [sq(Π 1 ), c) where sq(Π 1 ) is a certain uncountable cardinal that is smaller than c in some models of ZFC. In each non-discrete Polish Abelian group G we construct two closed subsets A,B ⊂ G with indP (A) = indP (B) = c and IndP (A ∪ B) = 1 and then apply this result to show that G contains a nowhere dense Haar null subset C ⊂ G with indP (C) = IndP (C) = κ for any given cardinal number κ ∈ [4, c].