On the Pair Correlation Density for Hyperbolic Angles

Dubi Kelmer, Alex Kontorovich

Published 2013-08-03, updated 2014-05-07Version 3

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.