Angles in hyperbolic lattices: The pair correlation density

Abstract

Abstract. It is well known that the angles in a lattice acting on hyperbolic nspace 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, Paşol, 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.

Directions in orbits of geometrically finite hyperbolic subgroups

“…Herein we will answer these questions and give a full characterisation of the spatial statistics of such a point set as viewed from a fixed observer in hyperbolic space or its boundary. These questions have been addressed previously for lattices [1,7,13,20], and for certain thin groups [26,27]. However we will treat a much more general class of subgroups in arbitrary dimension.…”

The gap distribution of directions in some Schottky groups

“…We let Ω n i,j,k (w, , l, s; T ) be the collection of (t, θ) which satisfies (14), (15), (16), (17), (18) and (20). It's clear that Ω n i,j,k (w, , l, s; T ) = 2 log T + Ω n i,j,k (w, , l, s; 0).…”

Directions in hyperbolic lattices

“…The recent work of Risager and Södergren [13] extends the effective convergence of the 2-point correlation function in [8] to arbitrary dimension n 2; it also includes an explicit formula for the limit in dimension n = 3. The analysis in [13] is restricted to the 2-point correlations of distances between the projected points on S n−1 . The approach presented here yields 2-point (as well as higher order) correlations of both distances and relative orientation, as we permit test sets that are not rotationally invariant.…”