Sumset phenomenon in countable amenable groups |
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Authors: | Mathias Beiglböck Vitaly Bergelson Alexander Fish |
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Institution: | a Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria b Department of Mathematics, Ohio State University, Columbus, OH 43210, United States |
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Abstract: | Jin proved that whenever A and B are sets of positive upper density in Z, A+B is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains Zd. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or — depending on the notation — “product sets”) are piecewise Bohr, a result which for G=Z was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group G, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density. |
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Keywords: | Amenable group Banach density Bohr set Piecewise syndetic Sumset phenomenon |
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