The horocycle flow at prime times

Abstract

We prove that the orbit of a non-periodic point at prime values of the horocycle flow in the modular surface is dense in a set of positive measure. For some special orbits we also prove that they are dense in the whole space—assuming the Ramanujan/Selberg Conjectures for GL2/Q. In the process, we derive an effective version of Dani's Theorem for the (discrete) horocycle flow.