Continued fractions and lines across the Stern--Brocot diagram

TL;DR

This paper explores how modifications in continued fraction expansions of rational numbers relate to geometric lines in the Stern-Brocot diagram, revealing a structured pattern of vertices on Euclidean lines and their convergence behavior.

Contribution

It establishes a geometric relationship between continued fractions and the Stern-Brocot diagram, showing vertices align on specific Euclidean lines and converge to a common point.

Findings

Vertices lie on two Euclidean lines crossing the diagram.

The lines' slopes differ only by a sign and meet at a specific point.

Vertices converge to this point as the parameter m tends to infinity.

Abstract

This paper concerns the relationships between continued fractions and the geometry of the Stern-Brocot diagram. Each rational number can be expressed as a continued fraction $[a_0; a_1, \ldots, a_n]$ whose terms $a_i$ are integers and are positive if $i \geq 1$. Select an index $i \in \{ 1, \ldots, n \}$ and replace $a_i$ with an integer $m$ to obtain a continued fraction expansion for an extended rational $\alpha_m \in \mathbb{Q} \cup \{ \infty \}$. This paper shows that the vertices of the Stern-Brocot diagram corresponding to the numbers $\{ \alpha_m \}_{m \in \mathbb{Z}}$ lie on a pair of (extended) Euclidean lines across the diagram. The slopes of these two lines differ only by a sign change and they meet at the point $L=\left([a_0; a_1, \ldots, a_{i-1}], 0\right) \in \mathbb{R}^2$. Moreover, as $\lvert m \rvert \to \infty$, the associated vertices move down these lines and converge to $L$. This paper concludes with a discussion which interprets this result in the context of 2-bridge link complements and Thurston's work on hyperbolic Dehn surgery.