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

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.

31 pages, 1 figure. Accepted by Transactions of the AMS. v3 (7-26-13) Title changed from 'An inverse theorem: when m(A+B)=m(A)+m(B) in a locally compact abelian group '. Many errors corrected, proofs clarified. The main argument is substantially shortened