On the Markoff spectrum on the Hecke group of index six

2025/06/10 by Byung-Chul Cha, Donghan Kim, Cha, Byungchul +3
Mathematics · #Mathematical Dynamics and Fractals #Advanced Algebra and Geometry #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2506.08358

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/√3. We show that both the Markoff and Lagrange spectra below 4/√(3) + ε have positive Hausdorff dimension for any positive ε. We also find maximal gaps and an isolated point in the spectra.

Citations

• The Markoff and Lagrange spectra on the Hecke group H4
• Attractors of dual continued fractions
• Lagrange spectrum of a circle over the Eisensteinian field
• Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum
• Geometric properties of the Markov and Lagrange spectra
• The dynamics of Pythagorean triples

Related

• The Markoff and Lagrange spectra on the Hecke group H4
• Markov spectrum near Freiman's isolated points in M∖ L
• On the graph of the dimension function of the Lagrange and Markov spectra
• Hausdorff dimension of Gauss--Cantor sets and two applications to classical Lagrange and Markov spectra
• Fractal geometry of the complement of Lagrange spectrum in Markov spectrum