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 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.