Limits of translates of divergent geodesics and Integral points on one-sheeted hyperboloids

TL;DR

This paper investigates the distribution of divergent geodesic translates in hyperbolic spaces and applies the results to count integral points on one-sheeted hyperboloids, revealing asymptotic formulas with logarithmic factors.

Contribution

It provides a detailed description of limit distributions of translates of divergent geodesics and derives asymptotic counts for integral points on hyperboloids with explicit error terms.

Findings

Limit distribution of orthogonal translates of divergent geodesics described.

Asymptotic count of lattice points on hyperboloids with logarithmic growth established.

Smoothed counts of integral quadratic forms with fixed discriminant derived.

Abstract

For any non-uniform lattice $\Gamma $ in $SL(2,R)$, we describe the limit distribution of orthogonal translates of a divergent geodesic in $\Gamma \backslash SL(2,R)$. As an application, for a quadratic form $Q$ of signature $(2,1)$, a lattice $\Gamma $ in its isometry group, and $v_0\in R^3$ with $Q(v_0)>0$, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit $v_0\Gamma $ of norm at most $T$, when the stabilizer of $v_0$ in $\Gamma $ is finite. Our result in particular implies that for any non-zero integer $d$, the smoothed count for number of integral binary quadratic forms with discriminant $d^2$ and with coefficients bounded by $T$ is asymptotic to $c\cdot T \log T +O(T)$.