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://www.numdam.org/articles/10.1016/j.crma.2014.12.007/

Bibliographie
Cité par

[1] Aka, M.; Breuillard, E.; Rosenzweig, L.; de Saxcé, N. 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.

[3] Bourgain, J.; Gamburd, A. On the spectral gap for finitely generated subgroups of $SU (2)$ , Invent. Math., Volume 171 (2008), pp. 83-121

[4] V. Beresnevich, D. Kleinbock, G. Margulis, Non-planarity and metric Diophantine approximation for systems of linear forms, J. Théor. Nombres Bordeaux, preprint , in press. | arXiv

[5] Dani, S.G. Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., Volume 359 (1985), pp. 55-89

[6] Gamburd, A.; Jakobson, D.; Sarnak, P. Spectra of elements in the group ring of $SU (2)$ , J. Eur. Math. Soc., Volume 1 (1999) no. 1, pp. 51-85

[7] Gorodnik, A. Open problems in dynamics and related fields, J. Mod. Dyn., Volume 1 (2007) no. 1, pp. 1-35

[8] Kleinbock, D. Extremal subspaces and their submanifolds, Geom. Funct. Anal., Volume 13 (2003) no. 2, pp. 437-466

[9] Kleinbock, Dmitry An extension of quantitative nondivergence and applications to Diophantine exponents, Trans. Amer. Math. Soc., Volume 360 (2008) no. 12, pp. 6497-6523

[10] Kleinbock, Dmitry An ‘almost all versus no’ dichotomy in homogeneous dynamics and Diophantine approximation, Geom. Dedic., Volume 149 (2010), pp. 205-218

[11] Kleinbock, D.Y.; Margulis, G.A. Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math., Volume 148 (1998) no. 1, pp. 339-360

[12] Kleinbock, D.; Margulis, G.; Wang, J. Metric Diophantine approximation for systems of linear forms via dynamics, Int. J. Number Theory, Volume 6 (2010) no. 5, pp. 1139-1168

[13] Margulis, G.A. The action of unipotent groups in a lattice space, Mat. Sb., Volume 86 (1971) no. 128, pp. 552-556

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