${\rm HD}(M\setminus L) > 0.353$
Volume 188 / 2019
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). From their works, we know 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 ${\rm HD}(M\setminus L)$ near $\alpha_{\infty}$.
A consequence of our results is the lower bound ${\rm HD}(M\setminus L) \gt 0.353$. 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).
