Central limit theorems for simultaneous Diophantine approximations

Dmitry Dolgopyat; Bassam Fayad; Ilya Vinogradov

doi:10.5802/jep.37

Central limit theorems for simultaneous Diophantine approximations
[Théorème central limite pour des approximations diophantiennes simultanées]

Dmitry Dolgopyat ¹ ; Bassam Fayad ² ; Ilya Vinogradov ³

¹ University of Maryland, Department of Mathematics 4176 Campus Dr., College Park, MD 20742-4015, USA
² Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot 58-56, avenue de France, Boite Courrier 7012, 75205 Paris Cedex 13, France
³ Princeton University, Department of Mathematics Fine Hall, Washington Rd., Princeton NJ 08544, USA

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 1-35.

Résumé
Abstract

Nous étudions la loi de probabilité modulo $1$ des valeurs prises sur les entiers par $r$ formes linéaires de $d$ variables à coefficients aléatoires. Nous montrons un théorème central limite, « en moyenne » et « presque sûr », pour le nombre de points atteignant simultanément des cibles de rayon décroissant à une vitesse $n^{- r / d}$ . D’après le théorème de Khintchine-Groshev sur les approximations diophantiennes, $r / d$ est le seuil critique à partir duquel le nombre des points tend vers l’infini.

We study the distribution modulo $1$ of the values taken on the integers of $r$ linear forms in $d$ variables with random coefficients. We obtain quenched and annealed central limit theorems for the number of simultaneous hits into shrinking targets of radii $n^{- r / d}$ . By the Khintchine-Groshev theorem on Diophantine approximations, $r / d$ is the critical exponent for the infinite number of hits.

Reçu le : 2016-04-12
Accepté le : 2016-10-29
Publié le : 2016-11-22

MR Zbl

DOI : 10.5802/jep.37

Classification : 60F05, 37A17, 11K60
Keywords: Central limit theorem, weakly dependent random variables, diophantine approximation, linear forms, space of lattices
Mot clés : Théorème central limite, variables aléatoires faiblement dépendantes, approximation diophantienne, formes linéaires, espace de réseaux

Affiliations des auteurs :

Dmitry Dolgopyat ¹ ; Bassam Fayad ² ; Ilya Vinogradov ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Dmitry Dolgopyat and Bassam Fayad and Ilya Vinogradov},
     title = {Central limit theorems for simultaneous {Diophantine} approximations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1--35},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.37},
     zbl = {1387.60046},
     mrnumber = {3583273},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.37/}
}

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Dmitry Dolgopyat; Bassam Fayad; Ilya Vinogradov. Central limit theorems for simultaneous Diophantine approximations. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 1-35. doi : 10.5802/jep.37. https://jep.centre-mersenne.org/articles/10.5802/jep.37/

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