In homogeneous dynamics, Ratner theory provides us with a classification of orbits of the unipotent flow, with important consequences in number theory as illustrated by Margulis’ (earlier) resolution of the Oppenheim conjecture. In more recent years, our attention has shifted to understand the asymptotic distribution of sparse sets along such orbits — here Ratner’s theory does not apply. I will report on joint work with Uri Shapira and Shucheng Yu in which we study a simple but important such model: evenly spaced points along an expanding closed horoycle on the modular surface.