Asymptotic formulae for the number of lattice points in Euclidean and Lobachevskij spaces. (English. Russian original) Zbl 0632.10048

Russ. Math. Surv. 42, No. 3, 13-42 (1987); translation from Usp. Mat. Nauk 42, No. 3(255), 13-38 (1987).
Let \(\Sigma\) be a Riemannian space with metric d and \(\Gamma\) a cofinite discontinuous group of motions of \(\Sigma\). Choose \(w,w_ 0\in \Sigma\) and denote by \(N(T;w_ 0,w)\) the number of elements \(\gamma\in \Gamma\) such that \(d(\gamma w,w_ 0)\leq T\). The author studies the asymptotic behaviour of \(N(T;w_ 0,w)\) as \(T\to \infty\) for Euclidean and for hyperbolic spaces. These lattice point problems were previously discussed by P. D. Lax and R. S. Phillips [J. Funct. Anal. 46, 280-350 (1982; Zbl 0497.30036)]. An important tool in the approach of Lax and Phillips is the principle of conservation of energy for the solutions of the wave equation.
The author suggests a different approach: He starts from a formula expanding \(N(T;w_ 0,w)\) in terms of the eigenfunctions of the Laplacian and takes the mean value with respect to one of the variables w, \(w_ 0\). Then he applies a well-known estimate for the spectral function of an elliptic operator and obtains the asymptotics for \(N(T;w_ 0,w)\) as \(T\to \infty\). The error terms are slightly better than those of Lax and Phillips. The author points out that his method applies equally well to groups with the finite geometric property instead of cofinite groups \(\Gamma\). This paper is well written.
Reviewer: J.Elstrodt


11P21 Lattice points in specified regions
11F27 Theta series; Weil representation; theta correspondences
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs


Zbl 0497.30036
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