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 :
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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
Dolgopyat, Dmitry 1 ; Fayad, Bassam 2 ; Vinogradov, Ilya 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
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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://archive.numdam.org/articles/10.5802/jep.37/

[1] Athreya, J. S.; Ghosh, A.; Tseng, J. Spiraling of approximations and spherical averages of Siegel transforms, J. London Math. Soc. (2), Volume 91 (2015) no. 2, pp. 383-404 | DOI | MR | Zbl

[2] Athreya, J. S.; Ghosh, A.; Tseng, J. Spherical averages of Siegel transforms for higher rank diagonal actions and applications (2014) (arXiv:1407.3573)

[3] Athreya, J. S.; Parrish, A.; Tseng, J. Ergodic theory and Diophantine approximation for linear forms and translation surfaces, Nonlinearity, Volume 29 (2016) no. 8, pp. 2173-2190 | DOI | Zbl

[4] Badziahin, D.; Beresnevich, V. V.; Velani, S. Inhomogeneous theory of dual Diophantine approximation on manifolds, Adv. Math., Volume 232 (2013), pp. 1-35 | DOI | MR | Zbl

[5] Beresnevich, V. V.; Bernik, V. I.; Kleinbock, D. Y.; Margulis, G. A. Metric Diophantine approximation: the Khintchine-Groshev theorem for nondegenerate manifolds, Moscow Math. J., Volume 2 (2002) no. 2, pp. 203-225 | DOI | MR | Zbl

[6] Beresnevich, V. V.; Velani, S. Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem, Internat. Math. Res. Notices (2010) no. 1, pp. 69-86 | MR | Zbl

[7] Cassels, J. W. S. An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, 45, Cambridge University Press, New York, 1957 | MR | Zbl

[8] Cornfeld, I. P.; Fomin, S. V.; Sinaĭ, Ya. G. Ergodic theory, Grundlehren Math. Wiss., 245, Springer-Verlag, New York, 1982 | MR | Zbl

[9] Dani, S. G. Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. reine angew. Math., Volume 359 (1985), pp. 55-89 Erratum: Ibid, 360 (1985), p. 214 | MR | Zbl

[10] Dolgopyat, D. Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc., Volume 356 (2004) no. 4, pp. 1637-1689 | DOI | MR | Zbl

[11] Dolgopyat, D.; Fayad, B. Deviations of ergodic sums for toral translations II. Boxes (2012) (arXiv:1211.4323)

[12] Duffin, R. J.; Schaeffer, A. C. Khintchine’s problem in metric Diophantine approximation, Duke Math. J., Volume 8 (1941), pp. 243-255 | DOI | MR | Zbl

[13] Edwards, S. The rate of mixing for diagonal flows on spaces of affine lattices (2013) (preprint, diva2:618047)

[14] El-Baz, D.; Marklof, J.; Vinogradov, I. The distribution of directions in an affine lattice: two-point correlations and mixed moments, Internat. Math. Res. Notices (2015) no. 5, pp. 1371-1400 | MR | Zbl

[15] Groshev, A. A theorem on a system of linear forms, Dokl. Akad. Nauk SSSR, Volume 19 (1938), pp. 151-152

[16] Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers, The Clarendon Press, Oxford University Press, New York, 1979 | Zbl

[17] Khintchine, A. Ein Satz über Kettenbrüche, mit arithmetischen Anwendungen, Math. Z., Volume 18 (1923), pp. 289-306 | DOI | Zbl

[18] Khintchine, A. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., Volume 92 (1924), pp. 115-125 | DOI | Zbl

[19] Kleinbock, D. Y.; Margulis, G. A. Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinaĭ’s Moscow Seminar on Dynamical Systems (Amer. Math. Soc. Transl. Ser. 2), Volume 171, American Mathematical Society, Providence, RI, 1996, pp. 141-172 | MR | Zbl

[20] Kleinbock, D. Y.; Margulis, G. A. Logarithm laws for flows on homogeneous spaces, Invent. Math., Volume 138 (1999) no. 3, pp. 451-494 | DOI | MR | Zbl

[21] Kleinbock, D. Y.; Shi, R.; Weiss, B. Pointwise equidistribution with an error rate and with respect to unbounded functions, Math. Ann. (2016) (doi:10.1007/s00208-016-1404-3, arXiv:1505.06717)

[22] Le Borgne, S. Principes d’invariance pour les flots diagonaux sur SL (d,)/ SL (d,), Ann. Inst. H. Poincaré Probab. Statist., Volume 38 (2002) no. 4, pp. 581-612 | DOI | MR | Zbl

[23] Marklof, J. The n-point correlations between values of a linear form, Ergodic Theory Dynam. Systems, Volume 20 (2000) no. 4, pp. 1127-1172 | DOI | MR

[24] Marklof, J. Distribution modulo one and Ratner’s theorem, Equidistribution in number theory, an introduction (NATO Sci. Ser. II Math. Phys. Chem.), Volume 237, Springer, Dordrecht, 2007, pp. 217-244 | DOI | MR | Zbl

[25] Philipp, W. Mixing sequences of random variables and probabilistic number theory, Mem. Amer. Math. Soc., 114, American Mathematical Society, Providence, RI, 1971 | Zbl

[26] Ratner, M. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel J. Math., Volume 16 (1973), pp. 181-197 | DOI | MR | Zbl

[27] Samur, J. D. A functional central limit theorem in Diophantine approximation, Proc. Amer. Math. Soc., Volume 111 (1991) no. 4, pp. 901-911 | DOI | MR | Zbl

[28] Schmidt, W. M. A metrical theorem in Diophantine approximation, Canad. J. Math., Volume 12 (1960), pp. 619-631 | DOI | MR | Zbl

[29] Schmidt, W. M. Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc., Volume 110 (1964), pp. 493-518 | DOI | MR | Zbl

[30] Schmidt, W. M. Badly approximable systems of linear forms, J. Number Theory, Volume 1 (1969), pp. 139-154 | DOI | MR | Zbl

[31] Sinaĭ, Ya. G. The central limit theorem for geodesic flows on manifolds of constant negative curvature, Soviet Math. Dokl., Volume 1 (1960), pp. 983-987 | MR | Zbl

[32] Strömbergsson, A. An effective Ratner equidistribution result for SL(2,) 2 , Duke Math. J., Volume 164 (2015) no. 5, pp. 843-902 | DOI | MR

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