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.

Rational points on nonlinear horocycles and pigeonhole statistics for the fractional parts of

“…As is shown in Proposition 3.1, moving from n(k/N)a(N) to n((k + 1)/N)a(N) corresponds, up to some negligible error, to right multiplication by the unipotent element u (1), where…”

“…distribution of this sequence in not Poissonian (see also Remark 1.6), which contrasts with the conjectured gap distribution of the fractional parts of n α for any other α ∈ (0, 1) \ { 1 2 }. In our case, if we instead considered the fractional parts of n α for α ∈ (0, 1) \ { 1 2 }, we would expect Poissonian pigeonhole statistics in the sense that the corresponding limiting distribution functions E j (s) would equal s j e −j /j !. This contrasts with the case α = 1 2 , as shown in Figure 1.…”

“…Based on their work, we would expect the functions E j (s) to be piecewise analytic with the functions becoming increasingly complex as j increases. for n ≤ N when α is equal to 1 2 , 1 3 and 2 3 . For α = 1/2, these proportions approximate E j (s) for s = 1.…”

“…The gap FIGURE 1. The proportion of N = 10 000 000 intervals in partition of T containing 0 ≤ j ≤ 6 points of n α + Z for n ≤ N when α is equal to 1 2 , 1 3 and 2 3 . For α = 1/2, these proportions approximate E j (s) for s = 1.…”

“…An effective equidistribution theorem for rational horocycle points in the modular surface is proved by Burrin, Shapira and Yu in [1, Theorem 1.1]. Using spectral methods, [1] shows such points equidistribute when the number of such rational points N being considered at height y satisfies for some . This contrasts with Theorem 1.10, which corresponds to the case when .…”