ISSN:
1432-1807
Keywords:
Mathematics Subject Classification (1991):11P21, (53C35, 58G25, 22E40)
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. We study the asymptotics of the lattice point counting function $N(x,y;r)=\#\{\gamma\in\Gamma\,:\,d(x,\gamma y)\}$ for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group $\Gamma$ of motions in X, such that $\Gamma\backslash X$ has finite volume. We show that \[ \displaystyle N(x,y;r) = \sum_{j=0}^m c_j \varphi_j(x) \varphi_j(y) e^{(\rho+\nu_j)r} + O_{x,y,\varepsilon}\left( e^{(2\rho n/(n+1) +\varepsilon)r}\right) \] as $r\rightarrow\infty$ , for each $\varepsilon〉0$ . The constant $2\rho$ corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions $\varphi_j\in L^2(\Gamma\backslash X)$ of the Laplacian, such that the eigenvalues $\rho^2-\nu_j^2$ are less than $4n\rho^2/(n+1)^2$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s002080050331
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