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arXiv · 2009.13608

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

Claire Burrin·Uri Shapira·Shucheng Yu
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Abstract

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.

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Claire Burrin, Uri Shapira, Shucheng Yu. 2020-09-28. Translates of rational points along expanding closed horocycles on the modular surface. https://doi.org/10.1007/s00208-021-02267-7

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