Shrinking target equidistribution of horocycles in cusps

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 $T\to \infty$. 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 $T/\ell \to 0$ 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.

32 pages. Minor changes from the previous version. This version is the accepted manuscript. To appear in Mathematische Zeitschrift