Uniform Diophantine approximation on the Hecke group $\mathbf H_4$

TL;DR

This work extends uniform Diophantine approximation to the Hecke group $\mathbf H_4$, proving a Dirichlet-type bound with universal constant $K=(\sqrt{2}+1)/2$ for all $\alpha\notin\mathbb Q(\mathbf H_4)$ and introducing $K(\alpha)$ as the best uniform-approximation constant. It shows that the $\mathbf H_4$-best approximations are exactly the convergents of the Rosen and dual Rosen continued fractions, linking geometric Ford-circle dynamics with symbolic $\mathbf H_4$-expansions via matrix products $A_d$ and $J$. The paper provides a Legendre-type refinement with an optimal bound $1/2$ for convergents and establishes precise relationships between Rosen dynamics and best approximations, yielding a complete description of uniform approximation on $\mathbf H_4$ and demonstrating sharpness through explicit examples. The results combine hyperbolic geometry, continued fraction theory, and matrix dynamics to advance the understanding of Diophantine approximation on Fuchsian groups beyond the modular case.

Abstract

Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound. We study uniform Diophantine approximation properties on the Hecke group $\mathbf H_4$. For a given real number $α$, we characterize the sequence of $\mathbf H_4$-best approximations of $α$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $α$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.

Figures (6)

• Figure 1: Fundamental domain of the Hecke group $\mathbf H_4$
• Figure 2: The fundamental domain $\bm\Omega$ of $\mathbb H_4$ (left) and the ideal quadrilateral $\bm\Delta$ (right)
• Figure 3: Cutting sequence along the geodesic from $\infty$ to $\alpha = [3,2,3,1,\dots]$
• Figure 4: Ford circles for the Hecke group $\mathbf H_4$
• Figure 5: The action of $G^{-1}_n$ on the upper half plane $\mathbb H$. The dashed circles are Ford circles based at 0 and infinity, and the dotted line is the geodesic with end points $-1$ and $1$.

Theorems & Definitions (39)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Example 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3