New gaps on the Lagrange and Markov spectra
Luke Jeffreys1 ; Carlos Matheus2 ; Carlos Gustavo Moreira3, 4
1 School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK
2 Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France
3 SUSTech International Center for Mathematics Shenzhen, Guangdong, People’s Republic of China
4 IMPA, Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, Brazil
Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 311-338.

On note respectivement L et M les spectres de Lagrange et de Markov. Il est connu que L⊂M et que M∖L≠⌀. Dans ce travail, on détecte de nouvelles lacunes dans L et M en utilisant les deux méthodes suivantes. Premièrement, on obtient de telles lacunes en décrivant une nouvelle partie de M∖L proche de 3,938 : cette région (avec trois autres candidats) a été trouvée en étudiant les images de L récemment produites par V. Delecroix et les deux derniers auteurs à l’aide de l’algorithme expliqué dans l’un des appendices de cet article. En outre, on obtient les plus grands éléments connus de M∖L et on améliore la minoration de la dimension de Hausdorff de M∖L obtenue par les deux derniers auteurs avec M. Pollicott et P. Vytnova (heuristiquement, on obtient une nouvelle minoration de la dimension de M∖L par 0,593). Deuxièmement, on utilise une idée de renormalisation et un critère d’épaisseur (issu de la thèse de doctorat du troisième auteur) pour détecter une infinité de lacunes maximales de M s’accumulant près de la lacune de Freiman précédant le célèbre rayon de Hall [4,52782956616⋯,∞) ⊂L.

Let L and M denote the Lagrange and Markov spectra, respectively. It is known that L⊂M and that M∖L≠⌀. In this work, we exhibit new gaps of L and M using two methods. First, we derive such gaps by describing a new portion of M∖L near to 3.938: this region (together with three other candidates) was found by investigating the pictures of L recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of M∖L and we improve upon a lower bound on the Hausdorff dimension of M∖L obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of 0.593 on the dimension of M∖L). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author’s PhD thesis) to detect infinitely many maximal gaps of M accumulating to Freiman’s gap preceding the so-called Hall’s ray [4.52782956616⋯,∞)⊂L.

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DOI : 10.5802/jtnb.1280
Classification : 11J06, 11A55
Mots clés : Lagrange and Markov spectra, maximal gaps, Hausdorff dimension
Luke Jeffreys 1 ; Carlos Matheus 2 ; Carlos Gustavo Moreira 3, 4

1 School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK
2 Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France
3 SUSTech International Center for Mathematics Shenzhen, Guangdong, People’s Republic of China
4 IMPA, Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, Brazil
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Luke Jeffreys; Carlos Matheus; Carlos Gustavo Moreira. New gaps on the Lagrange and Markov spectra. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 311-338. doi : 10.5802/jtnb.1280. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1280/

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