Moscow Mathematical Journal
Volume 21, Issue 4, October–December 2021 pp. 767–788.
M∖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>m1>m2>m3>m4 of M∖L lying in distinct connected components of R∖L, where M is the Markov spectrum and L is the Lagrange spectrum. These elements are part of a decreasing sequence (mk)k∈N of elements in M converging to 3 and we give some evidence towards the possibility that mk∈M∖L for all k≥1. In particular, this indicates that 3 might belong to the closure of M∖L. So, M∖L would not be closed near 3 and there would not exist ε>0 such that M∩(−∞,3+ε)=L∩(−∞,3+ε).
2020 Math. Subj. Class. 11A55, 11J06.
Keywords: Markov spectrum, Lagrange spectrum, Diophantine approximation.
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