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

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 $L$).