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 → ∞. We show that a closed horocycle whose length ` 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 /` → 0 and, for any δ > 0, the segment remains larger than max n T −1/6 ,(T /`) 1/2 o (T /`) −δ . We also have an effective result for a smaller range of rates of growth of T and `. Finally, a number-theoretic identity involving the Euler totient function follows from our technique.