The weighted Farey sequence and a sliding section for the horocycle flow

TL;DR

This paper extends the classical Farey sequence by introducing weights to denominators, characterizing the resulting gap distribution through a novel geometric approach involving horocycle flow and Poincaré sections.

Contribution

It provides a new characterization of weighted Farey sequence gap distributions using horocycle flow techniques, refining previous results with different methods.

Findings

Weighted Farey sequence gap distribution characterized as a scaled variable on a pentagon.

Construction of a time-varying Poincaré section for horocycle flow.

Explicit formulas for the limit transverse measure of the flow.

Abstract

The Farey sequence is the sequence of all rational numbers in the real unit interval, stratified by increasing denominators. A classical result by Hall says that its normalized gap distribution is the same as the distribution of the random variable 1/(2 zeta(2) xy) on a certain unit triangle. In this paper we weight the denominators by an arbitrary piecewise-smooth continuous function, and we characterize the resulting gap distribution as that of a multiple of the above variable, defined on a certain unit pentagon. Our characterization refines previous results by Boca, Cobeli and Zaharescu, but employs completely different techniques. Building upon recent work by Athreya and Cheung, we construct a varying-with-time Poincar\'e section for the horocycle flow on the space of unimodular lattices, and we interpret the weighted Farey sequence as the list of return times to the section. Under an appropriate parametrization, our pentagon appears as the orbit of Hall's triangle under the motion of the section, and basic equidistribution results for long closed horocycles yield explicit formulas for the limit transverse measure.