Angles in hyperbolic lattices : The pair correlation density

TL;DR

This paper derives a formula for the pair correlation density of angles in hyperbolic lattices across any dimension, extending previous 2D results and providing explicit asymptotic behaviors and effective estimates.

Contribution

It introduces a general formula for pair correlation density of angles in hyperbolic lattices in arbitrary dimensions, extending prior 2D results with explicit asymptotics and effective bounds.

Findings

Derived a formula for pair correlation density in hyperbolic lattices

Analyzed asymptotic behavior of the density function in small and large limits

Extended results from 2D to higher dimensions with explicit estimates

Abstract

It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we determine, among other things, the asymptotic behavior of the density function in both the small and large variable limits. This extends earlier results by Boca, Pasol, Popa and Zaharescu and Kelmer and Kontorovich in dimension 2 to general dimension $n$. Our proofs use the decay of matrix coefficients together with a number of careful estimates, and lead to effective results with explicit rates.