A reinforcement of the Bourgain-Kontorovich's theorem

Zaremba’s conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,\mathrm{.}\mathrm{.}\mathrm{.},d_k],$ with all partial quotients $d_1,d_2,\mathrm{.}\mathrm{.}\mathrm{.},d_k$ being bounded by an absolute constant $A\mathrm{.}$ 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 $\mathbb{N}\mathrm{.}$ In this paper the same theorem is proved with A=7.