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Equidistribution of primitive rational points on expanding horospheres

Published online by Cambridge University Press:  09 November 2015

Manfred Einsiedler ,
Shahar Mozes ,
Nimish Shah  and
Uri Shapira
Manfred Einsiedler
Affiliation:
ETH Zürich, CH-8092 Zürich, Switzerland email manfred.einsiedler@math.ethz.ch
Shahar Mozes
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, Givaat Ram, 91904 Jerusalem, Israel email mozes@math.huji.ac.il
Nimish Shah
Affiliation:
Ohio State University, Columbus 43210-1174, USA email shah@math.osu.edu
Uri Shapira
Affiliation:
Technion - Israel Institute of Technology, 32000 Haifa, Israel email ushapira@tx.technion.ac.il
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Abstract

We confirm a conjecture of Marklof regarding the limiting distribution of certain sparse collections of points on expanding horospheres. These collections are obtained by intersecting the expanded horosphere with a certain manifold of complementary dimension and turns out to be of arithmetic nature. This result is then used along the lines suggested by Marklof to give an analogue of a result of Schmidt regarding the distribution of shapes of lattices orthogonal to integer vectors.

Keywords

equidistributionhomogeneous spacesshapes of orthogonal lattices
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 4 , April 2016 , pp. 667 - 692
Copyright
© The Authors 2015 

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References

Aka, M., Einsiedler, M. and Shapira, U., Integer points on spheres and their orthogonal grids, J. Lond. Math. Soc. (2), to appear, arXiv:1411.1272.Google Scholar
Aka, M., Einsiedler, M. and Shapira, U., Integer points on spheres and their orthogonal lattices (with an appendix by Ruixiang Zhang), Preprint (2015), arXiv:1502.04209.Google Scholar
Clozel, L., Oh, H. and Ullmo, E., Hecke operators and equidistribution of Hecke points, Invent. Math. 144(2) (2001), 327–351; MR 1827734 (2002m:11044).CrossRefGoogle Scholar
Einsiedler, M., Ratner’s theorem on SL(2, ℝ)-invariant measures, Jahresber. Dtsch. Math.-Ver. 108(3) (2006), 143–164; MR 2265534 (2008b:37048).Google Scholar
Eskin, A. and Oh, H., Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems 26(1) (2006), 163–167; MR 2201942 (2006j:11068).Google Scholar
Glasner, E., Ergodic theory via joinings, Mathematical Surveys and Monographs, vol. 101 (American Mathematical Society, Providence, RI, 2003); MR 1958753 (2004c:37011).Google Scholar
Iwaniec, H., Spectral methods of automorphic forms, Graduate Studies in Mathematics, vol. 53, second edition (American Mathematical Society, Providence, RI, 2002); MR 1942691 (2003k:11085).Google Scholar
Kleinbock, D., Shah, N. and Starkov, A., Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook of dynamical systems, Vol. 1A (North-Holland, Amsterdam, 2002), 813–930; MR 1928528 (2004b:22021).Google Scholar
Margulis, G. A., On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics (Springer, Berlin, 2004); with a survey by Richard Sharp: periodic orbits of hyperbolic flows, translated from the Russian by Valentina Vladimirovna Szulikowska; MR 2035655 (2004m:37049).Google Scholar
Marklof, J., The asymptotic distribution of Frobenius numbers, Invent. Math. 181(1) (2010), 179–207; MR 2651383 (2011e:11133).Google Scholar
Marklof, J. and Strömbergsson, A., Diameters of random circulant graphs, Combinatorica 33(4) (2013), 429–466; MR 3133777.Google Scholar
Mozes, S. and Shah, N., On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems 15(1) (1995), 149–159.Google Scholar
Ratner, M., On Raghunathan’s measure conjecture, Ann. of Math. (2) 134(3) (1991), 545–607; MR 1135878 (93a:22009).CrossRefGoogle Scholar
Ratner, M., Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63(1) (1991), 235–280; MR 1106945 (93f:22012).Google Scholar
Schmidt, W. M., The distribution of sublattices of Zm, Monatsh. Math. 125(1) (1998), 37–81; MR 1485976 (99c:11083).CrossRefGoogle Scholar
Swinnerton-Dyer, H. P. F., A brief guide to algebraic number theory, London Mathematical Society Student Texts, vol. 50 (Cambridge University Press, Cambridge, 2001); MR 1826558 (2002a:11117).CrossRefGoogle Scholar
Shah, N. A., Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, in Lie groups and ergodic theory (Tata Institute for Fundamental Research, Mumbai, 1998), 229–271; MR 1699367 (2001a:22012).Google Scholar
Witte, D., Measurable quotients of unipotent translations on homogeneous spaces, Trans. Amer. Math. Soc. 345(2) (1994), 577–594; MR 1181187 (95a:22005).Google Scholar