Fine-scale distribution of roots of quadratic congruences

TL;DR

This paper investigates the fine-scale distribution of roots of quadratic congruences with square-free discriminants, using hyperbolic geometry to derive limit laws and explicit pair correlation densities.

Contribution

It introduces a geometric approach to analyze root distributions and provides explicit formulas for their pair correlation densities, advancing understanding of quadratic congruence roots.

Findings

Established limit laws for roots in small intervals

Derived explicit pair correlation density expressions

Connected root distribution to hyperbolic geodesic processes

Abstract

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence $\mu^2 \equiv D \bmod m$, with $D > 0$ square-free and $D\not\equiv 1 \bmod 4$. 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.