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 $$H\subseteq \mathrm {SL}(2,\mathbb {Z})$$ , we use a process developed by Fisher and Schmidt to lift a Poincare section of the horocycle flow on $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ found by Athreya and Cheung to the finite cover $$\mathrm {SL}(2,\mathbb {R})/H$$ of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ . 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, Szusz, and Turan.