Continued fractions and lines across the Stern--Brocot diagram

TL;DR

This work establishes a precise geometric relationship between perturbations of continued fraction expansions and the Stern-Brocot diagram. By using SL(2, Z) actions and Möbius transformations, it shows that for a standard continued fraction [a_0; a_1, ..., a_n] with a fixed i, the vertices ν(α_m) corresponding to α_m = [a_0; a_1, ..., a_{i-1}, m, a_{i+1}, ..., a_n] lie on two extended Euclidean lines ℓ^+ and ℓ^- that meet at L = ($[a_0; a_1, ..., a_{i-1}]$, 0), and converge to L as |m| → ∞. The proof reduces to a0 = 0 and expresses α_m as P(m)/Q(m) with Q(m) strictly increasing, showing ν(α_m) lies on one of the two lines according to the sign of Q(m); a Möbius conjugation handles delicate end cases. The discussion connects these results to the topology and geometry of 2-bridge links and Thurston’s hyperbolic Dehn surgery, interpreting the line-convergence in terms of hyperbolic link complements and their limits. The findings provide a coherent bridge between continued fraction combinatorics, Stern-Brocot geometry, and 3-manifold topology.

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 $α_m \in \mathbb{Q} \cup \{ \infty \}$. This paper shows that the vertices of the Stern-Brocot diagram corresponding to the numbers $\{ α_m \}_{m \in \mathbb{Z}}$ 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.

Figures (6)

• Figure 1: A portion of the Stern--Brocot diagram $\mathcal{G}$.
• Figure 2: The funnels $F_{-4/7}$ and $F_{2/7}$ correspond to the continued fraction expansions $-4/7=[-1;2,3]$ and $2/7=[0;3,2]$.
• Figure 3: Motivating cases \ref{['mcase 2']} and \ref{['mcase 3']} concern the numbers $[a_0; m]$ and $[a_0; 1, m]$. The corresponding Stern-Brocot vertices appear along Euclidean line segments as shown above.
• Figure 4: The vertices in $\mathcal{G}$ for $[0;3,m]=\frac{m}{1+3m}$ lie on the left edge of the triangle when $m\geq 0$. The vertices for $[0;2,1,m]=\frac{1+m}{2+3m}$ lie on the right edge when $m\geq 0$.
• Figure 5: This shows a portion of $\mathcal{G}$ along with the (blue) lines $\ell^\pm$ for $\alpha_m=[0;3,m,4]$. The vertices for $\nu(\alpha_m)$ are shown as pink dots. In this case, the lines $\ell^\pm$ are shared with those for the numbers $\beta_m=[0;2,1,m,4]$. The vertices for $\nu(\beta_m)$ are shown as yellow dots.

Theorems & Definitions (7)

• Theorem 1.1: Theorem 2.1 of Hat
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4