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We study the limiting distribution of the rational points under a horizontaltranslation along a sequence of expanding closed horocycles on the modularsurface. Using spectral methods we confirm equidistribution of these samplepoints for any translate when the sequence of horocycles expands within acertain polynomial range. We show that the equidistribution fails for generictranslates and a slightly faster expanding rate. We also prove bothequidistribution and non-equidistribution results by obtaining explicitlimiting measures while allowing the sequence of horocycles to expandarbitrarily fast. Similar results are also obtained for translates of primitiverational points.

Translates of rational points along expanding closed horocycles on the modular surface