Central limit theorems for simultaneous Diophantine approximations
[Théorème central limite pour des approximations diophantiennes simultanées]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 1-35.

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.

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DOI : https://doi.org/10.5802/jep.37
Classification : 60F05,  37A17,  11K60
Mots clés : Théorème central limite, variables aléatoires faiblement dépendantes, approximation diophantienne, formes linéaires, espace de réseaux
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     title = {Central limit theorems for simultaneous {Diophantine} approximations},
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Dolgopyat, Dmitry; Fayad, Bassam; Vinogradov, Ilya. Central limit theorems for simultaneous Diophantine approximations. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 1-35. doi : 10.5802/jep.37. http://www.numdam.org/articles/10.5802/jep.37/

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