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

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 L. In 2018 the first region of M L above 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 L above 12, in the vicinity of the Markov value of two periodic words of odd length that are non semisymmetric, which are 212332111 and 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 dH (M, L) (the Hausdorff distance between M and L).