  Zaremba's conjecture (1971) states that every positive integer number $d$ can
be represented as a denominator (continuant) of a finite continued fraction
$\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients
$d_1,d_2,...,d_{k}$ being bounded by an absolute constant $A.$ Recently (in
2011) several new theorems concerning this conjecture were proved by Bourgain
and Kontorovich. The easiest of them states that the set of numbers satisfying
Zaremba's conjecture with A=50 has positive proportion in $\N.$ In this paper
the same theorem is proved with A=7.
