# American Institute of Mathematical Sciences

2016, 10: 229-254. doi: 10.3934/jmd.2016.10.229

## Effective equidistribution of translates of maximal horospherical measures in the space of lattices

 1 Department of Mathematics, University of Texas, 1 University Station, Austin, TX 78712, United States 2 Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, United States 3 Department of Mathematics and Computer Science, Wesleyan University, 265 Church Street, Middletown, CT 06459, United States

Received  June 2015 Revised  February 2016 Published  July 2016

Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient conditions under which certain translates of homogeneous measures converge, and they determined the limiting measures in the cases of convergence. The class of measures they considered includes the maximal horospherical measures. In this paper we prove the corresponding effective equidistribution results in the space of unimodular lattices. We also prove the corresponding results for probability measures with absolutely continuous densities in rank two and three. Then we address the problem of determining the error terms in two counting problems also considered by Mohammadi and Salehi-Golsefidy. In the first problem, we determine an error term for counting the number of lifts of a closed horosphere from an irreducible, finite-volume quotient of the space of positive definite $n\times n$ matrices of determinant one that intersect a ball with large radius. In the second problem, we determine a logarithmic error term for the Manin conjecture of a flag variety over $\mathbb{Q}$.
Citation: Kathryn Dabbs, Michael Kelly, Han Li. Effective equidistribution of translates of maximal horospherical measures in the space of lattices. Journal of Modern Dynamics, 2016, 10: 229-254. doi: 10.3934/jmd.2016.10.229
##### References:
 [1] V. Bernik, D. Kleinbock and G. A. Margulis, Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions, Internat. Math. Res. Notices, (2001), 453-486. doi: 10.1155/S1073792801000241. [2] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math., 27 (1965), 55-150. [3] S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067. [4] W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179. doi: 10.1215/S0012-7094-93-07107-4. [5] A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209. doi: 10.1215/S0012-7094-93-07108-6. [6] J. Franke, Y. I. Manin and Y. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math., 95 (1989), 421-435. doi: 10.1007/BF01393904. [7] J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Springer-Verlag, New York-Heidelberg, 1975. [8] D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360. doi: 10.2307/120997. [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] D. Kleinbock, R. Shi and B. Weiss, Pointwise equidistribution with an error rate and with respect to unbounded functions, arXiv:1505.06717, 2015. [11] D. Kleinbock and B. Weiss, Dirichlet's theorem on Diophantine approximation and homogeneous flows, J. Mod. Dyn., 2 (2008), 43-62. [12] G. A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1. [13] A. Mohammadi and A. S. Golsefidy, Translate of horospheres and counting problems, Amer. J. Math., 136 (2014), 1301-1346. doi: 10.1353/ajm.2014.0037. [14] H. Oh, Orbital counting via mixing and unipotent flows, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 339-375. [15] P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602. [16] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), 20 (1956), 47-87. [17] N. Shah and B. Weiss, On actions of epimorphic subgroups on homogeneous spaces, Ergodic Theory Dynam. Systems, 20 (2000), 567-592. doi: 10.1017/S0143385700000298. [18] N. A. Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 106 (1996), 105-125. doi: 10.1007/BF02837164. [19] R. Shi, Expanding cone and applications to homogeneous dynamics, arXiv:1510.05256, 2015. [20] D. Zagier, Eisenstein series and the Riemann zeta-function, in Automorphic Forms, Representation Theory and Arithmetic, Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay, 275-301, 1981.

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##### References:
 [1] V. Bernik, D. Kleinbock and G. A. Margulis, Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions, Internat. Math. Res. Notices, (2001), 453-486. doi: 10.1155/S1073792801000241. [2] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math., 27 (1965), 55-150. [3] S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067. [4] W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179. doi: 10.1215/S0012-7094-93-07107-4. [5] A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209. doi: 10.1215/S0012-7094-93-07108-6. [6] J. Franke, Y. I. Manin and Y. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math., 95 (1989), 421-435. doi: 10.1007/BF01393904. [7] J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Springer-Verlag, New York-Heidelberg, 1975. [8] D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360. doi: 10.2307/120997. [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] D. Kleinbock, R. Shi and B. Weiss, Pointwise equidistribution with an error rate and with respect to unbounded functions, arXiv:1505.06717, 2015. [11] D. Kleinbock and B. Weiss, Dirichlet's theorem on Diophantine approximation and homogeneous flows, J. Mod. Dyn., 2 (2008), 43-62. [12] G. A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1. [13] A. Mohammadi and A. S. Golsefidy, Translate of horospheres and counting problems, Amer. J. Math., 136 (2014), 1301-1346. doi: 10.1353/ajm.2014.0037. [14] H. Oh, Orbital counting via mixing and unipotent flows, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 339-375. [15] P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602. [16] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), 20 (1956), 47-87. [17] N. Shah and B. Weiss, On actions of epimorphic subgroups on homogeneous spaces, Ergodic Theory Dynam. Systems, 20 (2000), 567-592. doi: 10.1017/S0143385700000298. [18] N. A. Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 106 (1996), 105-125. doi: 10.1007/BF02837164. [19] R. Shi, Expanding cone and applications to homogeneous dynamics, arXiv:1510.05256, 2015. [20] D. Zagier, Eisenstein series and the Riemann zeta-function, in Automorphic Forms, Representation Theory and Arithmetic, Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay, 275-301, 1981.
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