Sumset phenomenon in countable amenable groups

By M. Beiglboeck, Vitaly Bergelson, Alexander Fish. 2009

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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 Z d . 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 — “productsets”) 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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cfe8beb17c57f4d4d72232f0fe81f02c07cc6d42