Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Markoff spectrum on the Hecke group of index six

Byungchul Cha, Dong Han Kim, Deokwon Sim

TL;DR

This work analyzes the Lagrange and Markoff spectra for the index-$6$ Hecke group $\mathbf{H}_6$, focusing on the middle part beyond the smallest accumulation point $\frac{4}{\sqrt{3}}$. It develops the $\mathbf{H}_6$-expansion and a bi-infinite-sequence framework to compute spectra from symbolic data, enabling detailed gap analysis and dimensional results. The authors identify two explicit maximal gaps in the Markoff spectrum with endpoints $\frac{\sqrt{143}}{5}$, $\sqrt{7}$ and $\sqrt{7}$, $\frac{13\sqrt{3}+13\sqrt{7}+\sqrt{143}}{26}$, with $\sqrt{7}$ isolated, and prove that the spectra below $\frac{4}{\sqrt{3}}+\varepsilon$ have positive Hausdorff dimension. They also show the dimension persists near the accumulation point via an iterated function system, suggesting analogous behavior for higher-index Hecke groups. The work connects to Schmidt’s spectra and classical Markoff theory, extending the structural understanding of these Diophantine spectra to $\mathbf{H}_6$.

Abstract

The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A.~Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point $4/\sqrt3$. We show that both the Markoff and Lagrange spectra below $4/\sqrt{3} + ε$ have positive Hausdorff dimension for any positive $ε$. We also find maximal gaps and an isolated point in the spectra.

On the Markoff spectrum on the Hecke group of index six

TL;DR

This work analyzes the Lagrange and Markoff spectra for the index- Hecke group , focusing on the middle part beyond the smallest accumulation point . It develops the -expansion and a bi-infinite-sequence framework to compute spectra from symbolic data, enabling detailed gap analysis and dimensional results. The authors identify two explicit maximal gaps in the Markoff spectrum with endpoints , and , , with isolated, and prove that the spectra below have positive Hausdorff dimension. They also show the dimension persists near the accumulation point via an iterated function system, suggesting analogous behavior for higher-index Hecke groups. The work connects to Schmidt’s spectra and classical Markoff theory, extending the structural understanding of these Diophantine spectra to .

Abstract

The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A.~Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point . We show that both the Markoff and Lagrange spectra below have positive Hausdorff dimension for any positive . We also find maximal gaps and an isolated point in the spectra.

Paper Structure

This paper contains 7 sections, 19 theorems, 134 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Expansion of real numbers by the Hecke group
  3. Maximal gaps of the Markoff spectrum
  4. Computational preliminaries
  5. Proof of Theorem \ref{['thm:three_gaps1']}
  6. Proof of Theorem \ref{['thm:longest_gap']}
  7. Dimension of the Markoff spectrum

Key Result

Theorem 1.1

For any $\varepsilon >0$, we have

Figures (4)

  • Figure 1: Three closed geodesics in the modular surface$\mathbb{H}/\mathbf{H}_6$ with lowest heights.
  • Figure 2: Gaps in $\mathscr{M}(\mathbf{H}_6)$. (This figure is not drawn to scale.)
  • Figure 3: The fundamental domain $\Xi$ of the group $\mathbf H_6$ on the upper half plane (left) and the fundamental domain $\Sigma$ of the group $\bm\Gamma_6$ (right).
  • Figure 4: Maximal gaps and their boundary points in $\mathscr{M}(\mathbf{H}_6)$. (This figure is not drawn to scale.)

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • ...and 23 more