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

Griesmer, John

doi:10.1090/S0002-9947-2013-06022-9

An inverse theorem: When the measure of the sumset is the sum of the measures in a locally compact abelian group
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by John T. Griesmer PDF

Trans. Amer. Math. Soc. 366 (2014), 1797-1827 Request permission

Abstract:

We classify the pairs of subsets $A$, $B$ of a locally compact abelian group $G$ satisfying $m_*(A+B)=m(A)+m(B)$, where $m$ is the Haar measure for $G$ and $m_*$ is inner Haar measure. This generalizes M. Kneser’s classification of such pairs when $G$ is assumed to be connected. Recently, D. Grynkiewicz classified the pairs of sets $A$, $B$ satisfying $|A+B|=|A|+|B|$ in an abelian group, and our result is complementary to that classification. Our proofs combine arguments of Kneser and Grynkiewicz.

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