  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).
