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

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.

arXiv commentv2: 50 pages, minor revisions, updated references

References in corpus (6)

• The asymptotic distribution of Frobenius numbers
• Effective Limit Distribution of the Frobenius Numbers
• Equidistribution of primitive rational points on expanding horospheres
• Shrinking targets problems for flows on homogeneous spaces
• Primitive rational points on expanding horocycles in products of the modular surface with the torus
• Joint equidistribution on the product of the circle and the unit tangent bundle of the modular surface

Cited by in corpus (1)

• Joint equidistribution on the product of the circle and the unit tangent bundle of the modular surface