An inverse theorem: when the measure of the sumset is the sum of the measures in a locally compact abelian group

TL;DR

This paper characterizes pairs of subsets in locally compact abelian groups where the measure of their sumset equals the sum of their measures, extending Kneser's classification to a broader setting.

Contribution

It generalizes Kneser's theorem by classifying measure-equal sumset pairs in locally compact abelian groups, combining previous proofs with new arguments.

Findings

Classified measure-equal sumset pairs in locally compact abelian groups.

Extended Kneser's theorem beyond compact connected groups.

Unified measure and cardinality sumset classifications.

Abstract

We classify the pairs of subsets (A,B) of a locally compact abelian group satisfying m(A+B)=m(A)+m(B), where m is Haar measure. This generalizes a result of M. Kneser classifying such pairs under the additional assumption that G is compact and connected. Our proof combines Kneser's proof with arguments of D. Grynkiewicz, who classified the pairs of subsets (A,B) of abelian groups satisfying |A+B|=|A|+|B|, where |A| is the cardinality of A.