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 \(4n\rho^2/(n+1)^2\).
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Received: 4 January 1999
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Bruggeman, R., Miatello, R. & Wallach, N. Resolvent and lattice points on symmetric spaces of strictly negative curvature. Math Ann 315, 617–639 (1999). https://doi.org/10.1007/s002080050331
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DOI: https://doi.org/10.1007/s002080050331