New gaps on the Lagrange and Markov spectra

Jeffreys, Luke; Matheus, Carlos; Moreira, Carlos Gustavo

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 311-338

Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\ne ⌀$. 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 $M\setminus L$ 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 $M\setminus L$ and we improve upon a lower bound on the Hausdorff dimension of $M\setminus L$ 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 $M\setminus L$). 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 $[4.52782956616\cdots ,\infty )\subset L$.

On note respectivement $L$ et $M$ les spectres de Lagrange et de Markov. Il est connu que $L\subset M$ et que $M\setminus L\ne ⌀$. 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 $M\setminus L$ 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 $M\setminus L$ et on améliore la minoration de la dimension de Hausdorff de $M\setminus L$ obtenue par les deux derniers auteurs avec M. Pollicott et P. Vytnova (heuristiquement, on obtient une nouvelle minoration de la dimension de $M\setminus L$ 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 $[4,52782956616\cdots ,\infty )$ $\subset L$.

Classification : 11J06, 11A55
Keywords: Lagrange and Markov spectra, maximal gaps, Hausdorff dimension

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