Extensions of Schreiber’s theorem on discrete approximate subgroups in  d
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 149-162.

In this paper we give an alternative proof of Schreiber’s theorem which says that an infinite discrete approximate subgroup in d is relatively dense around a subspace. We also deduce from Schreiber’s theorem two new results. The first one says that any infinite discrete approximate subgroup in d is a restriction of a Meyer set to a thickening of a linear subspace in d , and the second one provides an extension of Schreiber’s theorem to the case of the Heisenberg group.

Dans cet article, nous donnons une autre démonstration du théorème de Schreiber : un sous-groupe approximatif discret infini de d est relativement dense au voisinage d’un sous-espace. Nous déduisons aussi du théorème de Schreiber deux nouveaux résultats : le premier affirme qu’un sous-groupe approximatif discret infini de d est la restriction d’un ensemble de Meyer à un épaississement d’un sous-espace linéaire de d , et le second propose une extension du théorème de Schreiber au cas du groupe de Heisenberg.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.90
Classification: 11B30,  52C23
Keywords: Approximate groups, approximate lattices, Meyer sets
Fish, Alexander 1

1 School of Mathematics and Statistics F07, University of Sydney NSW 2006, Australia
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Fish, Alexander. Extensions of Schreiber’s theorem on discrete approximate subgroups in $\protect \mathbb{R}^d$. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 149-162. doi : 10.5802/jep.90. http://www.numdam.org/articles/10.5802/jep.90/

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