Counting lattice points

Abstract

For a locally compact second countable group G and a lattice subgroup Gamma, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Gamma holds, with a rate of convergence. The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality. In addition, it holds uniformly over families of lattices satisfying a uniform spectral gap. Applications include counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties. The mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze-Stein phenomenon. We prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points.