A sparse equidistribution problem for expanding horocycles on the modular surface

The orbits of the horocycle flow on hyperbolic surfaces are classified: each orbit is either dense or a closed horocycle around a cusp. Expanding closed horocycles are themselves asymptotically dense, and in fact become equidistributed on the surface. The precise rate of equidistribution is of interest; on the modular surface, Zagier observed that a particular rate is equivalent to the Riemann hypothesis being true. I will discuss the asymptotic behavior of evenly spaced points along an expanding closed horocycle. This is based on joint work with Uri Shapira and Shucheng Yu.