Pair correlation of angles between reciprocal geodesics on the modular surface

Abstract

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

Pair correlation of hyperbolic lattice angles

“…Therefore it is more natural to consider the pair correlation measure R el Q defined as R Q , with the condition γ = γ ′ replaced by γω = γ ′ ω. Denoting by g el 2 the corresponding pair correlation functions, we have g el 2 (ξ) = g 2 (e ω ξ), where e ω is the cardinality of the stabilizer of ω, so that g el 2 = g 2 if ω is not an elliptic point. For ω = i, the function g el 2 is identical with the pair correlation function found in [5], but the formula here is entirely explicit for all ξ.…”

“…A first step in the study of the pair correlation of directions of hyperbolic lattice points was completed in [5], where we treated the case Γ = PSL 2 (Z) and ω = i, establishing a formula for the pair correlation density g 2 (ξ) that involves two terms. The first term is a series over the set of matrices M with nonnegative entries of an explicit function of ξ depending only on the Hilbert-Schmidt norm of M , while the second term is a finite sum involving volumes of bodies defined in terms of the triangle transformation introduced in [3].…”

Angles in hyperbolic lattices: The pair correlation density

“…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].…”

On the pair correlation density for hyperbolic angles

“…In particular, for the Euclidian analogue of this problem, the pair correlation, as well as other spacing statistics, were studied in [BCZ00, BZ05, BZ06, MS10, EMV13]. It is thus very surprising that the question of spacing statistics for the well-studied set of geodesic ray angles in the hyperbolic plane was considered for the first time only recently in [BPPZ12,BPZ13]. To state the results, we introduce some notation.…”

“…Remark 1.20. In [BPPZ12], another expression for g 2 is given, again for the case Γ = SL 2 (Z) and ω = i, in terms of lengths of reciprocal geodesics on the modular surface. More generally, keeping the base point ω = i, if we assume that Γ is invariant under transpose and there is another lattice Γ such that the matrices A = M t M with M ∈ Γ are all the symmetric matrices in Γ , then any sum over any function f ( (M )) with M ∈ Γ can be written as a sum over f ( (C)/2), where C runs over closed geodesics in Γ \H passing through i.…”