Extensions of Schreiber’s theorem on discrete approximate subgroups in $\protect \mathbb{R}^d$

Fish, Alexander

doi:10.5802/jep.90

Extensions of Schreiber’s theorem on discrete approximate subgroups in

ℝ^{d}

[Extensions du théorème de Schreiber sur les sous-groupes approximatifs discrets de

ℝ^{d}

]

Fish, Alexander ¹

¹ School of Mathematics and Statistics F07, University of Sydney NSW 2006, Australia

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 149-162.

Résumé
Abstract

Dans cet article, nous donnons une autre démonstration du théorème de Schreiber : un sous-groupe approximatif discret infini de $ℝ^{d}$ 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 $ℝ^{d}$ est la restriction d’un ensemble de Meyer à un épaississement d’un sous-espace linéaire de $ℝ^{d}$ , et le second propose une extension du théorème de Schreiber au cas du groupe de Heisenberg.

In this paper we give an alternative proof of Schreiber’s theorem which says that an infinite discrete approximate subgroup in $ℝ^{d}$ 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 $ℝ^{d}$ is a restriction of a Meyer set to a thickening of a linear subspace in $ℝ^{d}$ , and the second one provides an extension of Schreiber’s theorem to the case of the Heisenberg group.

Reçu le : 2018-02-06
Accepté le : 2019-01-09
Publié le : 2019-02-11

MR Zbl

DOI : 10.5802/jep.90

Classification : 11B30, 52C23
Keywords: Approximate groups, approximate lattices, Meyer sets
Mot clés : Groupes approximatifs, réseaux approximatifs, ensembles de Meyer

Affiliations des auteurs :

Fish, Alexander ¹

¹ School of Mathematics and Statistics F07, University of Sydney NSW 2006, Australia

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     title = {Extensions of {Schreiber{\textquoteright}s} theorem on discrete approximate subgroups in~$\protect \mathbb{R}^d$},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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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, Tome 6 (2019), pp. 149-162. doi : 10.5802/jep.90. http://www.numdam.org/articles/10.5802/jep.90/

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