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

Alexander Fish

doi:10.5802/jep.90

Extensions of Schreiber’s theorem on discrete approximate subgroups in

ℝ^{d}

Alexander Fish ¹

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

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

Abstract
Résumé

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.

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.

Received: 2018-02-05
Accepted: 2019-01-08
Published online: 2019-02-10

MR Zbl

DOI: 10.5802/jep.90

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

Author's affiliations:

Alexander Fish ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
     pages = {149--162},
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Alexander Fish. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.90/

References
Cited by

[1] M. Björklund & T. Hatnick - “Approximate lattices”, Duke Math. J. 167 (2018) no. 15, p. 2903-2964 | DOI | MR

[2] E. Breuillard, B. Green & T. Tao - “The structure of approximate groups”, Publ. Math. Inst. Hautes Études Sci. 116 (2012), p. 115-221 | DOI | MR

[3] G. A. Freĭman - Foundations of a structural theory of set addition, Translations of Mathematical Monographs, vol. 37, American Mathematical Society, Providence, R. I., 1973 | MR | Zbl

[4] B. Green & I. Z. Ruzsa - “Freiman’s theorem in an arbitrary abelian group”, J. London Math. Soc. (2) 75 (2007) no. 1, p. 163-175 | DOI | MR | Zbl

[5] E. Hrushovski - “Stable group theory and approximate subgroups”, J. Amer. Math. Soc. 25 (2012) no. 1, p. 189-243 | DOI | MR | Zbl

[6] J. C. Lagarias - “Meyer’s concept of quasicrystal and quasiregular sets”, Comm. Math. Phys. 179 (1996) no. 2, p. 365-376 | DOI | MR | Zbl

[7] Y. Meyer - Nombres de Pisot, nombres de Salem et analyse harmonique, Lect. Notes in Math., vol. 117, Springer-Verlag, Berlin-New York, 1970 | MR | Zbl

[8] R. V. Moody - “Model sets: survey” (2000), arXiv:math/0002020

[9] J.-P. Schreiber - Approximations diophantiennes et problèmes additifs dans les groupes abéliens localement compacts, Ph. D. Thesis, Université Paris-Sud, 1972

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