Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

Abstract

For a given finite index subgroup H ⊆ SL(2, Z), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on SL(2, R)/SL(2, Z) found by Athreya and Cheung to the finite cover SL(2, R)/H of SL(2, R)/SL(2, Z). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erdős, Szüsz, and Turán.

Equidistribution of Farey sequences on horospheres in covers of and applications

“…A,Q on T n ∆ so that for x ∈ T n ∆ , EST A A,c,Q (x) is the number of solutions (p, q) ∈ A satisfying (9), and K A A,Q (x) is the number of solutions satisfying (10). Our second main result is the following: Theorem 2.…”

“…In dimension n = 1, Athreya and Cheung [1] provided a unified explanation for various statistical properties of Farey fractions, which were originally proven by analytic methods, by realizing the horocycle flow in SL(2, R)/SL(2, Z) as a suspension flow over the BCZ map introduced by Boca, Cobeli, and Zaharescu [5] in their study of Farey fractions. The author's recent work [10] used a process of Fisher and Schmidt [9] to lift the Poincaré section of Athreya and Cheung to obtain sections of the horocycle flow in covers SL(2, R)/∆ of SL(2, R)/SL (2, Z), which in turn yielded results on the spacing statistics of the various subsets of Farey fractions related to those lifted sections.…”

The Erdős–Szüsz–Turán distribution for equivariant processes

“…We were introduced to this problem by Boca, Zaharescu, and Heersink (who has studied the problem of finding approximates with appropriate congruence conditions [16]). Their methods, number theoretic in nature, yield explicit formulas for f (A, c) (computed independently by Boca [8] and Xiong-Zaharescu [46]), and they also considered localizing α to smaller intervals.…”

The Boca-Cobeli-Zaharescu Map Analogue for the Hecke Triangle Groups $G_q$

“…Several other applications of the G3-BCZ map to the statistics of the visible lattice points Λ3 = Z 2 prim = {(x, y) ∈ Z 2 | gcd(x, y) = 1} can be similarly extended-almost verbatim-to general Λq. This list includes, but is not limited to, an old Diophantine approximation problem of [9] Erdös, P., Szüsz, P., & Turán solved independently by Xiong and Zaharescu [20], and Boca [5] for G3 = SL(2, Z), and Heersink [14] for finite index subgroups of G3 = SL(2, Z); the average depth of cusp excursions of the horocycle flow on X2 = SL(2, R)/G3 by Athreya and Cheung [3]; and the statistics of weighted Farey sequences by Panti [17].…”

“…In both cases, the sought for application was determining the slope gap distributions for the holonomy vectors of the golden L and the regular octagon. Soon after, B. Heersink [14] computed the BCZ map analogues for finite covers of SL(2, R)/ SL(2, Z) using a process developed by A. M. Fisher, and T. A. Schmidt [10] for lifting Poincaré sections of the geodesic flow on SL(2, R)/SL(2, Z) to covers of thereof.…”