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Dmitry Dolgopyat; Bassam Fayad; Ilya Vinogradov
Central limit theorems for simultaneous Diophantine approximations
(Théorème central limite pour des approximations diophantiennes simultanées)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 1-35, doi: 10.5802/jep.37
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Class. Math.: 60F05, 37A17, 11K60
Mots clés: Théorème central limite, variables aléatoires faiblement dépendantes, approximation diophantienne, formes linéaires, espace de réseaux

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.

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