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January  2018, 38(1): 155-168. doi: 10.3934/dcds.2018007

An application of lattice points counting to shrinking target problems

Dmitry Kleinbock , and  Xi Zhao

Brandeis University, Waltham MA, 02454, USA

* Corresponding author: Dmitry Kleinbock

Received  January 2017 Revised  July 2017 Published  September 2017

Fund Project: The first-named author was supported by NSF grants DMS-1101320 and DMS-1600814.

We apply lattice points counting results to solve a shrinking target problem in the setting of discrete time geodesic flows on hyperbolic manifolds of finite volume.

Keywords: Shrinking target problems, hyperbolic geometry, geodesic flows, counting of lattice points.
Mathematics Subject Classification: Primary:37D40;Secondary:53D25, 37A25.
Citation: Dmitry Kleinbock, Xi Zhao. An application of lattice points counting to shrinking target problems. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 155-168. doi: 10.3934/dcds.2018007
References:
[1]

J. Athreya, Logarithm laws and shrinking target properties, Proc. Indian Acad. (Math. Sci.), 119 (2009), 541-557.  doi: 10.1007/s12044-009-0044-x.  Google Scholar

[2]

_____, Cusp excursions on parameter spaces, J. Lond. Math. Soc. , 87 (2013), 741–765 Google Scholar

[3]

Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties, Annales de L'Institut Fourier, 62 (2012), 1889-1942.  doi: 10.5802/aif.2738.  Google Scholar

[4]

N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.  doi: 10.1007/BF02809888.  Google Scholar

[5]

D. Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc., 356 (2004), 1637-1689.  doi: 10.1090/S0002-9947-03-03335-X.  Google Scholar

[6]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/MRL.2007.v14.n5.a8.  Google Scholar

[7]

A. Gorodnik and A. Nevo, Counting lattice points, J. Reine Angew. Math., 663 (2012), 127-176.  doi: 10.1515/CRELLE.2011.096.  Google Scholar

[8]

A. Gorodnik and N. Shah, Khinchin's theorem for approximation by integral points on quadratic varieties, Math. Ann., 350 (2011), 357-380.  doi: 10.1007/s00208-010-0561-z.  Google Scholar

[9]

N. Haydn, M. Nicol, T. Persson and S. Vaienti, A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems, 33 (2013), 475-498.  doi: 10.1017/S014338571100099X.  Google Scholar

[10]

H. Huber, Über eine neue Klasse automorpher Functionen und eine Gitterpunktproblem in der hyperbolischen Ebene, Comment. Math. Helv., 30 (1956), 20-62.  doi: 10.1007/BF02564331.  Google Scholar

[11]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.  Google Scholar

[12]

P. Lax and R. Phillips, The asymptotic distribution of lattice points in Euclidean and Non-Euclidean spaces, J. Funct. Anal., 46 (1982), 280-350.  doi: 10.1016/0022-1236(82)90050-7.  Google Scholar

[13]

F. Maucourant, Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math., 152 (2006), 143-155.  doi: 10.1007/BF02771980.  Google Scholar

[14]

C. C. Moore, Exponential decay of correlation coefficients for geodesic flows, in: Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, CA, 1984), 163–181, Math. Sci. Res. Inst. Publ. 6, Springer, New York, 1987. doi: 10.1007/978-1-4612-4722-7_6.  Google Scholar

[15]

W. Philipp, Some metrical theorems in number theory, Pacific J. Math., 20 (1967), 109-127.  doi: 10.2140/pjm.1967.20.109.  Google Scholar

[16]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

[17]

D. Sullivan, Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.  doi: 10.1007/BF02392354.  Google Scholar

show all references

References:
[1]

J. Athreya, Logarithm laws and shrinking target properties, Proc. Indian Acad. (Math. Sci.), 119 (2009), 541-557.  doi: 10.1007/s12044-009-0044-x.  Google Scholar

[2]

_____, Cusp excursions on parameter spaces, J. Lond. Math. Soc. , 87 (2013), 741–765 Google Scholar

[3]

Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties, Annales de L'Institut Fourier, 62 (2012), 1889-1942.  doi: 10.5802/aif.2738.  Google Scholar

[4]

N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.  doi: 10.1007/BF02809888.  Google Scholar

[5]

D. Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc., 356 (2004), 1637-1689.  doi: 10.1090/S0002-9947-03-03335-X.  Google Scholar

[6]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/MRL.2007.v14.n5.a8.  Google Scholar

[7]

A. Gorodnik and A. Nevo, Counting lattice points, J. Reine Angew. Math., 663 (2012), 127-176.  doi: 10.1515/CRELLE.2011.096.  Google Scholar

[8]

A. Gorodnik and N. Shah, Khinchin's theorem for approximation by integral points on quadratic varieties, Math. Ann., 350 (2011), 357-380.  doi: 10.1007/s00208-010-0561-z.  Google Scholar

[9]

N. Haydn, M. Nicol, T. Persson and S. Vaienti, A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems, 33 (2013), 475-498.  doi: 10.1017/S014338571100099X.  Google Scholar

[10]

H. Huber, Über eine neue Klasse automorpher Functionen und eine Gitterpunktproblem in der hyperbolischen Ebene, Comment. Math. Helv., 30 (1956), 20-62.  doi: 10.1007/BF02564331.  Google Scholar

[11]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.  Google Scholar

[12]

P. Lax and R. Phillips, The asymptotic distribution of lattice points in Euclidean and Non-Euclidean spaces, J. Funct. Anal., 46 (1982), 280-350.  doi: 10.1016/0022-1236(82)90050-7.  Google Scholar

[13]

F. Maucourant, Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math., 152 (2006), 143-155.  doi: 10.1007/BF02771980.  Google Scholar

[14]

C. C. Moore, Exponential decay of correlation coefficients for geodesic flows, in: Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, CA, 1984), 163–181, Math. Sci. Res. Inst. Publ. 6, Springer, New York, 1987. doi: 10.1007/978-1-4612-4722-7_6.  Google Scholar

[15]

W. Philipp, Some metrical theorems in number theory, Pacific J. Math., 20 (1967), 109-127.  doi: 10.2140/pjm.1967.20.109.  Google Scholar

[16]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

[17]

D. Sullivan, Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.  doi: 10.1007/BF02392354.  Google Scholar

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