Poincar\'e sections for the horocycle flow in covers of SL(2,R)/SL(2,Z) and applications to Farey fraction statistics

Abstract

For a given finite index subgroup H of SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincar\'e 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 Erd\H{o}s, Sz\"usz, and Tur\'an.