Fine-scale distribution of roots of quadratic congruences

Abstract

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence µ 2 ≡ D mod m, with D > 0 square-free and D ≡ 1 mod 4. This is achieved by translating the problem to convergence of certain geodesic random line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.

A Poincaré map for the horocycle flow on $PSL(2,\mathbb{Z})\backslash \mathbb{H}$ and the Stern-Brocot tree

“…Let us remark that as far as we know the only other Poincaré maps for the horocycle flow on the modular surface which have been constructed are those in [4] (see also [13]) and in the recent [14]. These maps use a cross-section defined in terms of the identification of SH with the space of unimodular lattices.…”