Israel Journal of Mathematics

, Volume 176, Issue 1, pp 419–444

Strong wavefront lemma and counting lattice points in sectors

Article

Abstract

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.

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References

  1. [DRS]
    W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Mathematical Journal 71 (1993), 181–209.CrossRefMathSciNetGoogle Scholar
  2. [EM]
    A. Eskin and C. McMullen, Mixing, counting and equidistribution in Lie groups, Duke Mathematical Journal 71 (1993), 143–180.CrossRefMathSciNetGoogle Scholar
  3. [EMM]
    A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Annals of Mathematics (2) 147 (1998), 93–141.MATHCrossRefMathSciNetGoogle Scholar
  4. [EMS]
    A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Annals of Mathematics (2) 143 (1996), 253–299.MATHCrossRefMathSciNetGoogle Scholar
  5. [GO]
    A. Gorodnik and H. Oh, Orbits of discrete subgroups on a symmetric space and Furstenberg boundary, Duke Mathematical Joural 139 (2007), 483–525.MATHCrossRefMathSciNetGoogle Scholar
  6. [GOS]
    A. Gorodnik, H. Oh and N. Shah, Integral points on symmetric varieties and Satake compactifications, American Journal of Mathematics 131 (2009), 1–57.MATHMathSciNetGoogle Scholar
  7. [HS]
    G. Heckman and H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Perspectives in Mathematics, 16, Academic Press, New York, 1994.MATHGoogle Scholar
  8. [N]
    A. Nevo, Exponential volume growth, maximal functions on symmetric spaces, and ergodic theorems for semi-simple Lie groups, Ergodic Theory and Dynamical Systems 25 (2005), 1257–1294.MATHCrossRefMathSciNetGoogle Scholar
  9. [Sc]
    H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progress in Mathematics, 49, Birkhaüser Boston, Inc., Boston, MA, 1984.MATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Math DepartmentBrown UniversityProvidenceUSA
  3. 3.School of MathematicsTIFRMumbaiIndia

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