Counting lattice points in norm balls on higher rank simple Lie groups

TL;DR

This paper improves error estimates for counting lattice points in norm balls within higher rank simple Lie groups, especially for special linear groups, using refined spectral methods and spherical function bounds.

Contribution

It provides the first significant improvement over the longstanding bounds for lattice point counting in higher rank Lie groups, notably for special linear groups.

Findings

Error estimates are nearly twice as accurate as previous bounds.

Refined spectral estimates are effective for counting lattice points.

Detailed proofs for special linear groups are provided.

Abstract

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. We focus particularly on the case of the special linear groups where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly twice as good in some cases.