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

Abstract

The Farey sequence F(Q) at level Q is the sequence of irreducible fractions in [0, 1] with denominators not exceeding Q, arranged in increasing order of magnitude. A simple "next-term" algorithm exists for generating the elements of F(Q) in increasing or decreasing order. That algorithm, along with a number of other properties of the Farey sequence, was encoded by F. Boca, C. Cobeli, and A. Zaharescu into what is now known as the Boca-Cobeli-Zaharescu (BCZ) map, and used to attack several problems that can be described using the statistics of subsets of the Farey sequence. In this paper, we derive the Boca-Cobeli-Zaharescu map analogue for the discrete orbits Λq = Gq(1, 0) T of the linear action of the Hecke triangle groups Gq on the plane R 2 starting with a Stern-Brocot tree analogue for the said orbits (theorem 2.2). We derive the next-term algorithm for generating the elements of Λq in vertical strips in increasing order of slope, and present a number of applications to the statistics of Λq.

Farey Sequences for Thin Groups

“…Heersink [Hee17] generalized [Mar10] to certain congruence subgroups of Λ (still in the finite covolume setting). Furthermore, the method of [AC14] has been generalized to more general subgroups such as Hecke triangle groups (e.g [Tah19]). However we will not discuss this approach here.…”