Moscow Mathematical Journal

Volume 21, Issue 4, October–December 2021 pp. 767–788.

$\mathbb{M}\backslash \mathbb{L}$ Near 3

Authors: Davi Lima (1), Carlos Matheus (2), Carlos Gustavo Moreira (3), and Sandoel Vieira (3)
Author institution:(1) Instituto de Matemática, UFAL, Av. Lourival Melo Mota s/n, Maceio, Alagoas, Brazil
(2) CMLS, École Polytechnique, CNRS (UMR 7640), 91128, Palaiseau, France
(3) IMPA, Estrada Dona Castorina, 110. Rio de Janeiro, Rio de Janeiro-Brazil

Summary:

We construct four new elements $3.11>m_1>m_2>m_3>m_4$ of $\mathbb{M}\backslash \mathbb{L}$ lying in distinct connected components of $\mathbb{R}\setminus \mathbb{L}$, where $\mathbb{M}$ is the Markov spectrum and $\mathbb{L}$ is the Lagrange spectrum. These elements are part of a decreasing sequence $(m_k)_{k\in\mathbb{N}}$ of elements in $\mathbb{M}$ converging to $3$ and we give some evidence towards the possibility that $m_k\in \mathbb{M}\setminus \mathbb{L}$ for all $k\geq 1$. In particular, this indicates that $3$ might belong to the closure of $\mathbb{M}\setminus \mathbb{L}$. So, $\mathbb{M}\setminus \mathbb{L}$ would not be closed near $3$ and there would not exist $\varepsilon>0$ such that $\mathbb{M}\cap (-\infty,3+\varepsilon)=\mathbb{L}\cap (-\infty,3+\varepsilon)$.

2020 Math. Subj. Class. 11A55, 11J06.

Keywords: Markov spectrum, Lagrange spectrum, Diophantine approximation.

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