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New portions of $M\setminus L$ and a lower bound on the Hausdorff distance between $L$ and $M$

Clément Rieutord, Carlos Gustavo Moreira, Harold Erazo

Abstract

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 $M\setminus L\neq \emptyset$. 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 $L$).

New portions of $M\setminus L$ and a lower bound on the Hausdorff distance between $L$ and $M$

Abstract

Let and be the Markov and Lagrange spectra, respectively. It is known that is contained in and Freiman showed in 1968 that . In 2018 the first region of above 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 above , in the vicinity of the Markov value of two periodic words of odd length that are non semisymmetric, which are and . We will demonstrate that for both cases, there is a maximal gap of and a Gauss-Cantor set inside this gap that is contained in . Moreover we show that at the right endpoint of those gaps we have local Hausdorff dimension equal to . After studying the mentioned examples, we will provide a lower bound for the value of (the Hausdorff distance between and ).
Paper Structure (21 sections, 46 theorems, 256 equations, 2 figures)

This paper contains 21 sections, 46 theorems, 256 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Perron's description of the Lagrange and Markov spectra
  3. New portions of $M\setminus L$
  4. Hausdorff distance between $M$ and $L$
  5. Preliminaries
  6. Boundaries of finite words
  7. Forbidden words
  8. Portion of $M\setminus L$ in the vicinity of $\lambda_0(\overline{21233^*2111})$
  9. Local uniqueness
  10. Sequence development Tree around $j_0 = \lambda_0(\overline{21233^*2111})$
  11. Self-replication
  12. Focus on forbidden words with the smallest $\lambda_0$ value
  13. The description of $(M\setminus L)\cap(j_0,j_1)$
  14. The local border of $L$
  15. Portion of $M\setminus L$ in the vicinity of $\lambda_0(\overline{12333^*2112})$
  16. ...and 6 more sections

Key Result

Theorem 1.4

We define the Langrage and Markov value of a sequence $S \in (\mathbb{N}^*)^{\mathbb{Z}}$ by: We have:

Figures (2)

  • Figure 1: Sequence development Tree of $\overline{21233^*2111}$
  • Figure 2: Sequence development Tree of $\overline{12333^*2112}$

Theorems & Definitions (94)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Perron, 1921
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 84 more