  Freiman proved in 1968 that the Lagrange and Markov spectra do not coincide
by exhibiting a countable infinite collection $\mathcal{F}$ of isolated points
of the Markov spectrum which do not belong the Lagrange spectrum.
  In this paper, we describe the structure of the elements of the Markov
spectrum in the largest interval $(c_{\infty}, C_{\infty})$ containing
$\mathcal{F}$ and avoiding the Lagrange spectrum. In particular, we compute the
smallest known element $f$ of $M\setminus L$, and we show that the Hausdorff
dimension of the portion of the Markov spectrum between $c_{\infty}$ and
$C_{\infty}$ is $> 0.2628$.
