Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum

2023/09/26 by Moreira, Carlos Gustavo, Villamil, Christian Camilo Silva
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.14646

Abstract

We prove that for any η that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets k^-1((-∞,η]) and k^-1(η), which are the sets of irrational numbers with best constant of Diophantine approximation bounded by η and exactly η respectively, have the same Hausdorff dimension. We also show that, as η varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.

Related

• Concentration of dimension in the Lagrange spectrum
• Hausdorff dimension of some subsets of the Lagrange and Markov spectra near 3
• On the graph of the dimension function of the Lagrange and Markov spectra
• Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups
• On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set