2021/06/01 by Tseng, Jimmy
#11F03 #37D40 (Primary) 37F32 #41A60 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2106.00836
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 ℓ 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/ℓ → 0 and, for any δ>0, the segment remains larger than max\T-1/6,(T/ℓ)1/2\(T/ℓ)-δ. We also have an effective result for a smaller range of rates of growth of T and ℓ. Finally, a number-theoretic identity involving the Euler totient function follows from our technique.