  We study the limiting distribution of the rational points under a horizontal
translation along a sequence of expanding closed horocycles on the modular
surface. Using spectral methods we confirm equidistribution of these sample
points for any translate when the sequence of horocycles expands within a
certain polynomial range. We show that the equidistribution fails for generic
translates and a slightly faster expanding rate. We also prove both
equidistribution and non-equidistribution results by obtaining explicit
limiting measures while allowing the sequence of horocycles to expand
arbitrarily fast. Similar results are also obtained for translates of primitive
rational points.
