On the Pair Correlation Density for Hyperbolic Angles

TL;DR

This paper establishes the existence and provides an explicit formula for the pair correlation density of angles between geodesic rays in hyperbolic space, confirming a conjecture related to lattice point distributions.

Contribution

It proves the existence and derives an explicit formula for the pair correlation density of hyperbolic angles, advancing understanding of geometric lattice distributions.

Findings

Explicit pair correlation density formula derived.

Confirmed conjecture of Boca-Popa-Zaharescu.

Enhanced understanding of hyperbolic lattice angle distributions.

Abstract

Let $\Gamma< \mathrm{PSL}_2(\mathbb{R})$ be a lattice and $\omega\in \mathbb{H}$ a point in the upper half plane. We prove the existence and give an explicit formula for the pair correlation density function for the set of angles between geodesic rays of the lattice $\Gamma \omega$ intersected with increasingly large balls centered at $\omega$, thus proving a conjecture of Boca-Popa-Zaharescu.