Hyperbolic lattice point counting in unbounded rank

TL;DR

This paper develops a spectral analysis approach to count matrices in SL(n, Z) with bounded height, providing highly accurate asymptotic formulas that improve upon previous results for various ranks.

Contribution

It introduces a novel spectral analysis method for lattice point counting in unbounded rank, achieving significantly better error estimates.

Findings

Asymptotic formula for matrix count with strong error bounds

Effective for both small and large rank cases

Advances previous lattice point counting techniques

Abstract

We use spectral analysis to give an asymptotic formula for the number of matrices in SL(n, Z) of height at most T with strong error terms, far beyond the previous known, both for small and large rank.