# American Institute of Mathematical Sciences

2017, 24: 110-122. doi: 10.3934/era.2017.24.012

## Central limit theorems in the geometry of numbers

 1 Department of Mathematics, Chalmers, Gothenburg, Sweden 2 University of Bristol, Bristol, UK

Received  June 28, 2017 Published  October 2017

We investigate in this paper the distribution of the discrepancy of various lattice counting functions. In particular, we prove that the number of lattice points contained in certain domains defined by products of linear forms satisfies a Central Limit Theorem. Furthermore, we show that the Central Limit Theorem holds for the number of rational approximants for weighted Diophantine approximation in $\mathbb{R}^d$. Our arguments exploit chaotic properties of the Cartan flow on the space of lattices.

Citation: Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012
##### References:
 [1] J. Athreya, A. Ghosh and J. Tseng, Spiraling of approximations and spherical averages of Siegel transforms, J. Lond. Math. Soc., 91 (2015), 383-404.  doi: 10.1112/jlms/jdu082. [2] P. Billingsley, Probability and Measure, Third edition, Wiley Series in Probability and Mathematical Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1995. [3] M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative multiple mixing, submitted, arXiv: 1701.00945 [4] M. Björklund and A. Gorodnik, Central limit theorem for group actions which are exponentially mixing of all orders, submitted. [5] D. Dolgopyat, B. Fayad and I. Vinogradov, Central limit theorems for simultaneous Diophantine approximations, J. Éc. polytech. Math., 4 (2017), 1-36.  doi: 10.5802/jep.37. [6] M. Fréchet and J. Shohat, A proof of the generalized second-limit theorem in the theory of probability, Trans. Amer. Math. Soc., 33 (1931), 533-543.  doi: 10.1090/S0002-9947-1931-1501604-6. [7] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8. [8] M. Fuchs, On a problem of W. J. LeVeque concerning metric Diophantine approximation, Trans. Amer. Math. Soc., 355 (2003), 1787-1801.  doi: 10.1090/S0002-9947-02-03225-7. [9] D. Y. Kleinbock and G. A. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, New York, 2012,385-396. doi: 10.1007/978-1-4614-1260-1_18. [10] W. J. Leveque, On the frequency of small fractional parts in certain real sequences Ⅰ, Trans. Amer. Math. Soc., 87 (1958), 237-261.  doi: 10.2307/1993099. [11] W. J. Leveque, On the frequency of small fractional parts in certain real sequences Ⅱ, Trans. Amer. Math. Soc., 94 (1959), 130-149.  doi: 10.1090/S0002-9947-1960-0121350-1. [12] W. Philipp, Mixing Sequences of Random Variables and Probabilistic Number Theory, , Memoirs of the American Mathematical Society, No. 114, American Mathematical Society, Providence, R. I., 1971. [13] C. A. Rogers, Mean values over the space of lattices, Acta Math., 94 (1955), 249-287.  doi: 10.1007/BF02392493. [14] W. M. Schmidt, A metrical theorem in Diophantine approximation, Canad. J. Math., 12 (1960), 619-631.  doi: 10.4153/CJM-1960-056-0. [15] W. M. Schmidt, A metrical theorem in geometry of numbers, Trans. Amer. Math. Soc., 95 (1960), 516-529.  doi: 10.1090/S0002-9947-1960-0117222-9. [16] T. P. Speed, Cumulants and partition lattices, Austral. J. Statist., 25 (1983), 378-388.  doi: 10.1111/j.1467-842X.1983.tb00391.x.

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##### References:
 [1] J. Athreya, A. Ghosh and J. Tseng, Spiraling of approximations and spherical averages of Siegel transforms, J. Lond. Math. Soc., 91 (2015), 383-404.  doi: 10.1112/jlms/jdu082. [2] P. Billingsley, Probability and Measure, Third edition, Wiley Series in Probability and Mathematical Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1995. [3] M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative multiple mixing, submitted, arXiv: 1701.00945 [4] M. Björklund and A. Gorodnik, Central limit theorem for group actions which are exponentially mixing of all orders, submitted. [5] D. Dolgopyat, B. Fayad and I. Vinogradov, Central limit theorems for simultaneous Diophantine approximations, J. Éc. polytech. Math., 4 (2017), 1-36.  doi: 10.5802/jep.37. [6] M. Fréchet and J. Shohat, A proof of the generalized second-limit theorem in the theory of probability, Trans. Amer. Math. Soc., 33 (1931), 533-543.  doi: 10.1090/S0002-9947-1931-1501604-6. [7] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8. [8] M. Fuchs, On a problem of W. J. LeVeque concerning metric Diophantine approximation, Trans. Amer. Math. Soc., 355 (2003), 1787-1801.  doi: 10.1090/S0002-9947-02-03225-7. [9] D. Y. Kleinbock and G. A. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, New York, 2012,385-396. doi: 10.1007/978-1-4614-1260-1_18. [10] W. J. Leveque, On the frequency of small fractional parts in certain real sequences Ⅰ, Trans. Amer. Math. Soc., 87 (1958), 237-261.  doi: 10.2307/1993099. [11] W. J. Leveque, On the frequency of small fractional parts in certain real sequences Ⅱ, Trans. Amer. Math. Soc., 94 (1959), 130-149.  doi: 10.1090/S0002-9947-1960-0121350-1. [12] W. Philipp, Mixing Sequences of Random Variables and Probabilistic Number Theory, , Memoirs of the American Mathematical Society, No. 114, American Mathematical Society, Providence, R. I., 1971. [13] C. A. Rogers, Mean values over the space of lattices, Acta Math., 94 (1955), 249-287.  doi: 10.1007/BF02392493. [14] W. M. Schmidt, A metrical theorem in Diophantine approximation, Canad. J. Math., 12 (1960), 619-631.  doi: 10.4153/CJM-1960-056-0. [15] W. M. Schmidt, A metrical theorem in geometry of numbers, Trans. Amer. Math. Soc., 95 (1960), 516-529.  doi: 10.1090/S0002-9947-1960-0117222-9. [16] T. P. Speed, Cumulants and partition lattices, Austral. J. Statist., 25 (1983), 378-388.  doi: 10.1111/j.1467-842X.1983.tb00391.x.
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