On the pair correlation density for hyperbolic angles

Abstract

Let $\Gamma <{PSL}_2\left(\mathbb{R}\right)$ be a lattice and let $\omega \in \mathbb{H}$ be 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 by Boca, Popa, and Zaharescu.