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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: John T. Griesmer
Journal: Trans. Amer. Math. Soc. 366 (2014), 1797-1827
MSC (2010): Primary 11P70
Published electronically: December 13, 2013
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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 $ \vert A+B\vert=\vert A\vert+\vert B\vert$ in an abelian group, and our result is complementary to that classification. Our proofs combine arguments of Kneser and Grynkiewicz.


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Additional Information

John T. Griesmer
Affiliation: Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
Address at time of publication: Department of Mathematics, University of Denver, John Greene Hall, Room 203, 2360 S. Gaylord Street, Denver, Colorado 80208
Email: John.Griesmer@du.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06022-9
PII: S 0002-9947(2013)06022-9
Received by editor(s): April 14, 2012
Published electronically: December 13, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.