A note on the reinforcement of the Bourgain-Kontorovich's theorem

Abstract: 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.