  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.
