On metric Diophantine approximation in matrices and Lie groups

Aka, Menny; Breuillard, Emmanuel; Rosenzweig, Lior; de Saxcé, Nicolas

doi:10.1016/j.crma.2014.12.007

Number theory/Dynamical systems

On metric Diophantine approximation in matrices and Lie groups
[Approximation diophantienne métrique dans les matrices et les groupes de Lie]

Présenté par : Étienne Ghys

Aka, Menny ¹ ; Breuillard, Emmanuel ² ; Rosenzweig, Lior ³ ; de Saxcé, Nicolas ⁴

¹ Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
² Laboratoire de mathématiques, bâtiment 425, Université Paris-Sud (Paris-11), 91405 Orsay cedex, France
³ Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden
⁴ LAGA, Institut Galilée, Université Paris-13, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 185-189.

Résumé
Abstract

Nous étudions l'exposant diophantien des sous-variétés analytiques de matrices réelles $m \times n$ et répondons à certaines questions posées par Beresnevich, Kleinbock et Margulis. Nous identifions une famille d'obstructions algébriques à l'extrémalité d'une telle sous-variété, et donnons une formule pour l'exposant lorsque celle-ci est définie sur $Q$ . Enfin, nous appliquons ces résultats à la détermination de l'exposant diophantien des groupes de Lie nilpotents rationnels.

We study the Diophantine exponent of analytic submanifolds of $m \times n$ real matrices, answering questions of Beresnevich, Kleinbock, and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over $Q$ . We then apply these results to the determination of the Diophantine exponent of rational nilpotent Lie groups.

Reçu le : 2014-11-01
Accepté le : 2014-12-10
Publié le : 2015-01-28

DOI : 10.1016/j.crma.2014.12.007

Affiliations des auteurs :

Aka, Menny ¹ ; Breuillard, Emmanuel ² ; Rosenzweig, Lior ³ ; de Saxcé, Nicolas ⁴

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Aka, Menny; Breuillard, Emmanuel; Rosenzweig, Lior; de Saxcé, Nicolas. On metric Diophantine approximation in matrices and Lie groups. Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 185-189. doi : 10.1016/j.crma.2014.12.007. http://archive.numdam.org/articles/10.1016/j.crma.2014.12.007/

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