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.
Highlights
Let {Sn}n∈N be a sequence of “nice” subsets that become equidistributed in their ambient space
In this paper, generalizing the setting of [8], we study the equidistribution problem for the sets of rational and primitive rational points under an arbitrary horizontal translation x ∈ R/Z along a given sequence of expanding closed horocycles on M
The rest of our results deal with sequences {yn}n∈N that can decay arbitrarily fast, and give both positive and negative results. This is the main novelty of this paper; the handling of cases in which the sample points can be arbitrarily sparse on the closed horocycles they lie on
Summary
Let {Sn}n∈N be a sequence of “nice” subsets that become equidistributed in their ambient space. Various sparse equidistribution results have been obtained for expanding horospheres in the space of lattices SLn(R)/ SLn(Z) for n ≥ 3 [7,9,22,23,26] and in Hilbert modular surfaces [24] For each of these equidistribution results, assumptions on the expanding rate of the sequence {Sn}n∈N are crucial; the discrete subsets {Rn}n∈N lying on {Sn}n∈N can not be too sparse. It is not hard to see that if one flips the quantifiers, for any fixed horizontal translation x, there are sequences {yn}n∈N (approaching zero rapidly) such that equidistribution fails. The main novel result of this paper (Theorem 1.5) says that there are sequences {yn}n∈N approaching zero arbitrarily fast such that for almost every horizontal translation x the normalized counting measures Rn(x, yn) and its primitive counterpart do not equidistribute. The subsections describe more precisely the setting and results obtained
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