Small product sets in compact groups

Abstract

This paper addresses the structure of pairs (A B) of Borel sets in a compact and second countable group G with Haar probability measure mG which have positive mG-measures and satisfy the identity mG(AB) = mG(A) + mG(B) < When G is a compact, abelian and connected group, M. Kneser provided a very satisfactory description of such pairs, which roughly says that both A and B must be co-null subsets of pre-images of two closed intervals under the same surjective homomorphism onto the circle group. Much more recently, Griesmer was able to divide the set of all possible such pairs in any compact and abelian group into four (not completely disjoint) subclasses. Motivated by some applications to small product sets in countable amenable groups with respect to the left upper Banach density, we continue this study, and offer a description of pairs of Borel sets as above in any compact and second countable group with an abelian identity component. The relevance of these compact groups to the study of product sets in countable amenable groups will be explained. The main result of this paper asserts that in this non-abelian situation, essentially only one new class of examples of pairs as above can occur. Unfortunately, due to a wide variety of technicalities we will have to spend a non-trivial part of the introduction to carefully set up the tools and concepts necessary for the formulation of the main results and to prepare for their proofs.