Directions in hyperbolic lattices

TL;DR

This paper investigates the fine-scale distribution of lattice points in hyperbolic space, revealing new limit laws and connecting to recent results on lattice correlations in hyperbolic geometry.

Contribution

It introduces a new approach to analyze the fine-scale statistics of hyperbolic lattice projections using random hyperbolic lattices, extending prior work on 2-point correlations.

Findings

Established limit distributions for projected hyperbolic lattice points.

Connected the results to recent studies on 2-point correlations in hyperbolic lattices.

Provided a new perspective on the statistical behavior of hyperbolic lattice orbits.

Abstract

It is well known that the orbit of a lattice in hyperbolic $n$-space is uniformly distributed when projected radially onto the unit sphere. In the present work, we consider the fine-scale statistics of the projected lattice points, and express the limit distributions in terms of random hyperbolic lattices. This provides in particular a new perspective on recent results by Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by Kelmer and Kontorovich for general lattices in dimension $n=2$.