Let $M$ and $L$ be the Markov and Lagrange spectra, respectively. It is known that $L$ is contained in $M$ and Freiman showed in 1968 that $M\setminus L\ne \emptyset $. In 2018 the first region of $M\setminus L$ above $\sqrt{12}$ was discovered by C. Matheus and C. G. Moreira, thus disproving a conjecture of Cusick of 1975. In 2022, the same authors together with L. Jeffreys discovered a new region near 3.938. In this paper, we will study two new regions of $M\setminus L$ above $\sqrt{12}$, in the vicinity of the Markov value of two periodic words of odd length that are non semisymmetric, which are $\overline{212332111}$ and $\overline{123332112}$. We will demonstrate that for both cases, there is a maximal gap of $L$ and a Gauss–Cantor set inside this gap that is contained in $M$. Moreover we show that at the right endpoint of those gaps we have local Hausdorff dimension equal to $1$.
After studying the mentioned examples, we will provide a lower bound for the value of $d_H(M,L)$ (the Hausdorff distance between $M$ and $L$).
Soient $M$ et $L$ les spectres de Markov et de Lagrange respectivement. Il est connu que $L$ est contenu dans $M$, et Freiman démontra en 1968 que $M\setminus L\ne \emptyset $. En 2018, la première région de $M\setminus L$ fut découverte par C. Matheus et C. G. Moreira, réfutant ainsi une conjecture faite par Cusick en 1975. En 2022, les mêmes auteurs, avec L. Jeffreys, découvrirent une nouvelle région près de 3.938. Dans ce papier, nous étudions deux nouvelles régions de $M\setminus L$ au-dessus de $\sqrt{12}$, au voisinage de la valeur de Markov de deux mots périodiques non semi-symétriques de longueurs impaires, à savoir $\overline{212332111}$ et $\overline{123332112}$. Dans les deux cas, nous démontrons qu’il existe une lacune maximale de $L$ et un ensemble de Gauss–Cantor à l’intérieur de cette lacune qui est contenu dans $M$. En outre, nous montrons que la dimension de Hausdorff locale à l’extrémité droite de ces lacunes est égale à 1. Après avoir étudié ces exemples, nous donnerons une minoration pour la distance de Hausdorff $d_H(M,L)$ entre $M$ et $L$.
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Keywords: Lagrange and Markov spectra, Hausdorff dimension, Hausdorff distance.
Clément Rieutord 1 ; Carlos Gustavo Moreira 2, 3 ; Harold Erazo 3
CC-BY-ND 4.0
@article{JTNB_2025__37_2_599_0,
author = {Cl\'ement Rieutord and Carlos Gustavo Moreira and Harold Erazo},
title = {New portions of $M\setminus L$ and a lower bound on the {Hausdorff} distance between $L$ and $M$},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {599--646},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {2},
doi = {10.5802/jtnb.1334},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1334/}
}
TY - JOUR AU - Clément Rieutord AU - Carlos Gustavo Moreira AU - Harold Erazo TI - New portions of $M\setminus L$ and a lower bound on the Hausdorff distance between $L$ and $M$ JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 599 EP - 646 VL - 37 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1334/ DO - 10.5802/jtnb.1334 LA - en ID - JTNB_2025__37_2_599_0 ER -
%0 Journal Article %A Clément Rieutord %A Carlos Gustavo Moreira %A Harold Erazo %T New portions of $M\setminus L$ and a lower bound on the Hausdorff distance between $L$ and $M$ %J Journal de théorie des nombres de Bordeaux %D 2025 %P 599-646 %V 37 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1334/ %R 10.5802/jtnb.1334 %G en %F JTNB_2025__37_2_599_0
Clément Rieutord; Carlos Gustavo Moreira; Harold Erazo. New portions of $M\setminus L$ and a lower bound on the Hausdorff distance between $L$ and $M$. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 599-646. doi: 10.5802/jtnb.1334
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