Sumset phenomenon in countable amenable groups. (English) Zbl 1187.43002
R. Jin [Proc. Am. Math. Soc. 130, No. 3, 855–861 (2002; Zbl 0985.03066)] proved that whenever \(A\) and \(B\) are sets of positive upper density in \(\mathbb{Z}\), \(A+ B\) is piecewise syndetic. Jin’s theorem was subsequently generalized by R. Jin and H. J. Keisler [Trans. Am. Math. Soc. 355, No. 1, 79–97 (2003; Zbl 1047.03051)] to Abelian groups with some extra structure. Answering a question of Jin and Keisler, it is shown in the paper under review that this result holds true in amenable groups, that is, in the most general setting where it can be naturally formulated.
Sharpening previously known results, the authors also prove that \(A+ B\) is piecewise Bohr, i.e. contains arbitrarily long intervals from an open set in the Bohr topology. (In the integer case this is due to V. Bergelson, H. Furstenberg and B. Weiss [Piecewise-Bohr sets of integers and combinatorial number theory. Klazar, Martin (ed.) et al., Topics in discrete mathematics. Dedicated to Jarik Nešetřil on the occasion of his 60th birthday. Berlin: Springer, Algorithms and Combinatorics 26, 13–37 (2006; Zbl 1114.37008)].) It is shown that this provides characterization of sumsets in the commutative case: a subset of an Abelian group \(G\) is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.
The proofs combine combinatorial techniques with ergodic theoretic methods and the theory of almost periodic functions.
A particularly short proof of Jin’s original result is obtained through a “union version” of Furstenberg’s celebrated correspondence principle.
Sharpening previously known results, the authors also prove that \(A+ B\) is piecewise Bohr, i.e. contains arbitrarily long intervals from an open set in the Bohr topology. (In the integer case this is due to V. Bergelson, H. Furstenberg and B. Weiss [Piecewise-Bohr sets of integers and combinatorial number theory. Klazar, Martin (ed.) et al., Topics in discrete mathematics. Dedicated to Jarik Nešetřil on the occasion of his 60th birthday. Berlin: Springer, Algorithms and Combinatorics 26, 13–37 (2006; Zbl 1114.37008)].) It is shown that this provides characterization of sumsets in the commutative case: a subset of an Abelian group \(G\) is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.
The proofs combine combinatorial techniques with ergodic theoretic methods and the theory of almost periodic functions.
A particularly short proof of Jin’s original result is obtained through a “union version” of Furstenberg’s celebrated correspondence principle.
Reviewer: Harald Rindler (Wien)
MSC:
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 11B05 | Density, gaps, topology |
| 22B99 | Locally compact abelian groups (LCA groups) |
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