Constructing universally small subsets of a given packing index in Polish groups

Abstract

∈ A. So, being 1-almost disjoint is equivalent to being disjoint.A subset A ⊂ G with small packing index can be thought as large in a geometric sense becausein this case the group G does not contain many disjoint translation copies of A. It is natural tocompare this largeness property with other largeness properties that have topological or measure-theoretic nature. It turns out that a set of a group A can have small packing index (so can belarge in geometric sense) and simultaneously be small in other senses. In [3] it was proved that eachuncountable Polish Abelian group G contains a closed subset A ⊂ G that has large packing indexPack(A) = 1 but is nowhere dense and Haar null in G. According to Theorem 16.3 [9], under CH (theContinuum Hypothesis), each Polish group G contains a subset A with packing index pack(A) = 1,which is universally null in the sense that A has measure zero with respect to any atomless Borelprobability measure on G. In this paper we move further in this direction and prove that under