Descriptive Complexity of the Sizes of Subsets of Groups

Abstract

We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-Čech compactification βG of G, we prove, in particular, that the closure of the minimal ideal of βG is of type F σδ .