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.
Similar content being viewed by others
References
W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Mathematical Journal 71 (1993), 181–209.
A. Eskin and C. McMullen, Mixing, counting and equidistribution in Lie groups, Duke Mathematical Journal 71 (1993), 143–180.
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.
A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Annals of Mathematics (2) 143 (1996), 253–299.
A. Gorodnik and H. Oh, Orbits of discrete subgroups on a symmetric space and Furstenberg boundary, Duke Mathematical Joural 139 (2007), 483–525.
A. Gorodnik, H. Oh and N. Shah, Integral points on symmetric varieties and Satake compactifications, American Journal of Mathematics 131 (2009), 1–57.
G. Heckman and H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Perspectives in Mathematics, 16, Academic Press, New York, 1994.
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.
H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progress in Mathematics, 49, Birkhaüser Boston, Inc., Boston, MA, 1984.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first and the second authors partially supported by NSF 0400631 and NSF 0333397, respectively.
Rights and permissions
About this article
Cite this article
Gorodnik, A., Oh, H. & Shah, N. Strong wavefront lemma and counting lattice points in sectors. Isr. J. Math. 176, 419–444 (2010). https://doi.org/10.1007/s11856-010-0035-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-010-0035-8