The worst approximable rational numbers

TL;DR

This paper classifies and enumerates rational numbers with high approximation constants using hyperbolic geometry, simplifying previous symbolic dynamics approaches and distinguishing two types of worst approximable rationals.

Contribution

It introduces a geometric framework to classify worst approximable rationals, avoiding complex symbolic dynamics and clarifying their geometric nature.

Findings

Classifies all rationals with approximation constant ≥ 1/3

Identifies two types of worst approximable rationals: simple and non-simple geodesics

Uses hyperbolic geometry to simplify the understanding of approximation constants

Abstract

We classify and enumerate all rational numbers with approximation constant at least $\frac{1}{3}$ using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.