  Zaremba's conjecture (1971) states that every positive integer number $d$ can
be represented as a denominator (continuant) of a finite continued fraction
$\frac{b}{d}=[d_1,d_2,...,d_{k}],$ whose partial quotients $d_1,d_2,...,d_{k}$
belong to a finite alphabet $\A\subseteq\N.$ In this paper it is proved for an
alphabet $\A,$ such that the Hausdorff dimension $\delta_{\A}$ of the set of
infinite continued fractions whose partial quotients belong to $\A,$ that the
set of numbers $d,$ satisfying Zaremba's conjecture with the alphabet $\A,$ has
positive proportion in $\N.$ The result improves our previous reinforcement of
the corresponding Bourgain-Kontorovich's theorem.
