Comptes Rendus
Number theory/Dynamical systems
On metric Diophantine approximation in matrices and Lie groups
[Approximation diophantienne métrique dans les matrices et les groupes de Lie]
Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 185-189.

Nous étudions l'exposant diophantien des sous-variétés analytiques de matrices réelles m×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×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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.12.007

Menny Aka 1 ; Emmanuel Breuillard 2 ; Lior Rosenzweig 3 ; Nicolas de Saxcé 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
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     title = {On metric {Diophantine} approximation in matrices and {Lie} groups},
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Menny Aka; Emmanuel Breuillard; Lior Rosenzweig; Nicolas de Saxcé. On metric Diophantine approximation in matrices and Lie groups. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 185-189. doi : 10.1016/j.crma.2014.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.12.007/

[1] M. Aka; E. Breuillard; L. Rosenzweig; N. de Saxcé Diophantine properties of nilpotent Lie groups, Compos. Math. (2015) (32 pages. Published online: 13 January 2015) | DOI

[2] M. Aka, E. Breuillard, L. Rosenzweig, N. de Saxcé, Metric Diophantine approximation on matrices and Lie groups, in preparation.

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