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Comptes Rendus Mathématique
Volume 355, n° 8
pages 835-846 (août 2017)
Doi : 10.1016/j.crma.2017.07.007
Received : 10 May 2017 ;  accepted : 11 July 2017
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
 

Tushar Das a , Lior Fishman b , David Simmons c , Mariusz Urbański b
a University of Wisconsin – La Crosse, Department of Mathematics & Statistics, 1725 State Street, La Crosse, WI 54601, USA 
b University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA 
c University of York, Department of Mathematics, Heslington, York YO10 5DD, UK 

Abstract

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   matrices are both equal to  , 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.

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Résumé

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   sont toutes deux égales à  , 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.

The full text of this article is available in PDF format.
1  Although Khintchine's 1926 paper [[10]] includes a proof of the existence of   and   matrices possessing a certain property which implies that they are singular, it does not include a definition of singularity nor discuss any property equivalent to singularity.
2  For results considering the superlevelset, see Theorem 2.9.
3  This is of course unlike real gravity, which imposes an energy cost variable with respect to distance.


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