  This paper is dedicated to the study of two famous subsets of the real line,
namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem
2.1, provides a rigorous estimate on the smallest value $t_1$ such that the
portion of the Markov spectrum $(-\infty,t_1)\cap M$ has Hausdorff dimension
$1$. Our second result, Theorem 3.1, gives a new upper bound on the Hausdorff
dimension of the set difference $M\setminus L$.
  Our method combines new facts about the structure of the classical spectra
together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets
of continued fraction expansions whose entries satisfy appropriate
restrictions.
