Illinois Journal of Mathematics

Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups

Mauro Di Nasso and Martino Lupini

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Beiglböck, Bergelson and Fish proved that if subsets $A$, $B$ of a countable discrete amenable group $G$ have positive Banach densities $\alpha$ and $\beta$ respectively, then the product set $AB$ is piecewise syndetic, that is, there exists $k$ such that the union of $k$-many left translates of $AB$ is thick. Using nonstandard analysis, we give a shorter alternative proof of this result that does not require $G$ to be countable and moreover yields the explicit bound $k\le1/\alpha\beta$. We also prove with similar methods that if $\{A_{i}\}_{i=1}^{n}$ are finitely many subsets of $G$ having positive Banach densities $\alpha_{i}$ and $G$ is countable, then there exists a subset $B$ whose Banach density is at least $\prod_{i=1}^{n}\alpha_{i}$ and such that $BB^{-1}\subseteq\bigcap_{i=1}^{n}A_{i}A_{i}^{-1}$. In particular, the latter set is piecewise Bohr.

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Illinois J. Math., Volume 58, Number 1 (2014), 11-25.

First available in Project Euclid: 1 April 2015

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Primary: 03H05: Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] 43A07: Means on groups, semigroups, etc.; amenable groups 11B05: Density, gaps, topology 11B13: Additive bases, including sumsets [See also 05B10]


Di Nasso, Mauro; Lupini, Martino. Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups. Illinois J. Math. 58 (2014), no. 1, 11--25. doi:10.1215/ijm/1427897166.

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