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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Accepté le :
Publié le :
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
@article{JEP_2017__4__1_0,
     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},
     mrnumber = {3583273},
     zbl = {1387.60046},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.37/}
}
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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