Central limit theorems for simultaneous Diophantine approximations

Dolgopyat, Dmitry; Fayad, Bassam; Vinogradov, Ilya

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.

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

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