# $HD(M\setminus L)>0.353$

## TL;DR

This paper investigates the structure and Hausdorff dimension of the complement of the Lagrange spectrum in the Markov spectrum, providing new lower bounds and explicit elements, including the largest known member of this set.

## Contribution

It describes the local structure of $M\setminus L$ near a non-isolated point and constructs a Cantor set with matching Hausdorff dimension, establishing a lower bound of 0.353 for the Hausdorff dimension of $M\setminus L$.

## Key findings

- Hausdorff dimension of $M\setminus L$ exceeds 0.353
- Constructed a Cantor set with dimension matching $M\setminus L$ near a special point
- Explicitly identified new elements of $M\setminus L$, including its largest known member

## Abstract

The complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ was studied by many authors (including Freiman, Berstein, Cusick and Flahive). After their works, we disposed of a countable collection of points in $M\setminus L$. In this article, we describe the structure of $M\setminus L$ near a non-isolated point $\alpha_{\infty}$ found by Freiman in 1973, and we use this description to exhibit a concrete Cantor set $X$ whose Hausdorff dimension coincides with the Hausdorff dimension of $M\setminus L$ near $\alpha_{\infty}$. A consequence of our results is the lower bound $HD(M\setminus L)>0.353$ on the Hausdorff dimension $HD(M\setminus L)$ of $M\setminus L$. Another by-product of our analysis is the explicit construction of new elements of $M\setminus L$, including its largest known member $c\in M\setminus L$ (surpassing the former largest known number $\alpha_4\in M\setminus L$ obtained by Cusick and Flahive in 1989).