Shrinking target equidistribution of horocycles in cusps

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 $T\to \mathrm{\infty }$. We show that a closed horocycle whose length $\mathrm{\ell }$ 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 $T\mathrm{/}\mathrm{\ell }\to 0$ and, for any $\delta >0$, the segment remains larger than $\max \{T^{-1\mathrm{/}6},{\left(T\mathrm{/}\mathrm{\ell }\right)}^{1\mathrm{/}2}\}{\left(T\mathrm{/}\mathrm{\ell }\right)}^{-\delta }$. We also have an effective result for a smaller range of rates of growth of $T$ and $\mathrm{\ell }$. Finally, a number-theoretic identity involving the Euler totient function follows from our technique.