Comptes Rendus
no. 8
Number theory/Dynamical systems
A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation
[Un principe variationnel en géométrie paramétrique des nombres, illustré par des applications à la théorie métrique de l'approximation diophantienne]

Présenté par : the Editorial Board

Tushar Das1 ; Lior Fishman2 ; David Simmons3 ; Mariusz Urbański2
1 University of Wisconsin – La Crosse, Department of Mathematics & Statistics, 1725 State Street, La Crosse, WI 54601, USA
2 University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA
3 University of York, Department of Mathematics, Heslington, York YO10 5DD, UK
Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 835-846.

Nous établissons un nouveau lien entre la théorie métrique de l'approximation diophantienne et la géométrie paramétrique des nombres, en démontrant un principe variationnel permettant le calcul des dimensions de Hausdorff et d'entassement de nombreux ensembles d'intérêt en approximation diophantienne. Comme cas particulier, nous démontrons que les dimensions de Hausdorff et d'entassement de l'ensemble des matrices singulières de dimensions m×n sont toutes deux égales à mn(1−1m+n), démontrant ainsi une conjecture de Kadyrov, Kleinbock, Lindenstrauss et Margulis, et répondant par là même à une question soulevée par Bugeaud, Cheung et Chevallier. D'autres exemples d'application incluent le calcul des dimensions des ensembles de points satisfaisant des conjectures énoncées par Starkov et Schmidt.

We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular m×n matrices are both equal to mn(1−1m+n), thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis as well as answering a question of Bugeaud, Cheung, and Chevallier. Other applications include computing the dimensions of the sets of points witnessing conjectures of Starkov and Schmidt.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.07.007
Tushar Das 1 ; Lior Fishman 2 ; David Simmons 3 ; Mariusz Urbański 2

1 University of Wisconsin – La Crosse, Department of Mathematics & Statistics, 1725 State Street, La Crosse, WI 54601, USA
2 University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA
3 University of York, Department of Mathematics, Heslington, York YO10 5DD, UK 
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     journal = {Comptes Rendus. Math\'ematique},
     pages = {835--846},
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Tushar Das; Lior Fishman; David Simmons; Mariusz Urbański. A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 835-846. doi : 10.1016/j.crma.2017.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.07.007/

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