On the pair correlation density for hyperbolic angles

Abstract

Let Γ < PSL 2 (R) be a lattice and ω ∈ 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 Γω intersected with increasingly large balls centered at ω, thus proving a conjecture of Boca-Popa-Zaharescu.

Angles in hyperbolic lattices: The pair correlation density

“…We refer to Theorem 6.1 for details. For n = 2 our convergence rate is identical to that proved in [15].…”

“…More refined angle statistics has been studied only recently: Boca, Paşol, Popa and Zaharescu [1,3] studied the pair correlation statistics for hyperbolic angles when Γ = PSL 2 (Z) and z 0 = i or z 0 = e iπ/3 , and gave conjectures for all lattices Γ ⊆ PSL 2 (R) ∼ = SO 0 (2, 1) and all base points z 0 . These conjectures were later resolved by Kelmer and Kontorovich [15].…”

The gap distribution of directions in some Schottky groups

“…As a subgroup of a thin group Γ 0 , Γ is also thin. If Γ is of finite covolume, or a lattice in SL(2, R), Kelmer and Kontorovich [9] proved the limiting pair correlation of the directions of a single Γ -orbit in D. This generalizes the work of Boca, Popa, and Zaharescu [3] which deals with the case when Γ is SL(2, Z) and the observer is placed at an elliptic point. Later, the work [9] was further generalized by Risager and Södergren [16] to cover the cases SO(n, 1) with explicit convergence rate, and by Marklof and Vinogradov [12] which determines a large class of local statistics, including gap distribution.…”

Statistical Regularity of Apollonian Gaskets

“…In macroscopic physics, cosmologists use pair correlations to study the distribution of stars and galaxies. Within mathematics, there is also a rich literature on the spatial statistics of point processes arising from various settings, such as Riemann zeta zeros [24], fractional parts of { √ n, n ∈ Z + } [14], directions of lattice points [9], [8], [18], [27], [21], [13], Farey sequences and their generalizations [17], [7], [5], [29], [4], [2], and translation surfaces [1], [3], [33]. Our list of interesting works here is far from inclusive.…”