Fractal dimensions of the Markov and Lagrange spectra near 3

2022/08/31 by Harold Erazo, Carlos Gustavo Moreira, Erazo, Harold +5 · 2 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2208.14830

openalex publication_date 2022/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Lagrange spectrum L and Markov spectrum M are subsets of the real line with complicated fractal properties that appear naturally in the study of Diophantine approximations. It is known that the Hausdorff dimension of the intersection of these sets with any half-line coincide, that is, dim_H(L ∩ (-∞, t)) = dim_H(M ∩ (-∞, t)):= d(t) for every t ≥ 0. It is also known that d(3)=0 and d(3+ε)>0 for every ε>0. We show that, for sufficiently small values of ε > 0, one has the approximation d(3+ε) = 2⋅\fracW(e^c₀|log ε|)|log ε|+O((log |log ε|)/(|log ε|²)), where W denotes the Lambert function (the inverse of f(x)=xe^x) and c₀=-loglog((3+√(5))/2) ≈ 0.0383. We also show that this result is optimal for the approximation of d(3+ε) by "reasonable" functions, in the sense that, if F(t) is a C² function such that d(3+ε) = F(ε) + o((log |log ε|)/(|log ε|²)), then its second derivative F''(t) changes sign infinitely many times as t approaches 0.

Cited by

• Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3
• On irrationals with Lagrange value exactly 3

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