It is well known that on any given hyperbolic surface of finite area, a closed horocycle of length ℓ becomes asymptotically equidistributed as ℓ→∞. In this paper we prove that any subsegment of length greater than ℓ1/2 + ε of such a closed horocycle also becomes equidistributed as ℓ→∞. The exponent 1/2 + ε is the best possible and improves upon a recent result by Hejhal [He3]. We give two proofs of the above result; our second proof leads to explicit information on the rate of convergence. We also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to ℓ.
Andreas Strömbergsson. "On the uniform equidistribution of long closed horocycles." Duke Math. J. 123 (3) 507 - 547, 15 June 2004. https://doi.org/10.1215/S0012-7094-04-12334-6