Shrinking target equidistribution of horocycles in cusps

TL;DR

This paper studies the equidistribution of horocycles near cusps in hyperbolic surfaces, establishing conditions under which they become uniformly distributed in shrinking neighborhoods, with effective bounds and a number-theoretic application.

Contribution

It provides new conditions for horocycle equidistribution in cusps, including effective bounds and a novel number-theoretic identity derived from the technique.

Findings

Horocycles become equidistributed in shrinking neighborhoods under specified conditions.

Effective bounds are established for the rates of growth of T and bb.

A number-theoretic identity involving Euler's totient function is derived.

Abstract

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of $T^{-1}$ as $T \rightarrow \infty$. We show that a closed horocycle whose length $\ell$ goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate $T^{-1}$ provided that $T/\ell \rightarrow 0$ and, for any $\delta>0$, the segment remains larger than $\max\left\{T^{-1/6},\left(T/\ell\right)^{1/2}\right\}\left(T/\ell\right)^{-\delta}$. We also have an effective result for a smaller range of rates of growth of $T$ and $\ell$. Finally, a number-theoretic identity involving the Euler totient function follows from our technique.