A strengthening of a theorem of Bourgain–Kontorovich II

Zaremba’s conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac bd= [d_1, d_2 ,\dots , d_k]$, with all partial quotients $d_1, d_2 ,\dots , d_k$ 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 $\mathbb{N}$. In 2013 we proved this result with $A = 7$. In this paper the same theorem is proved with $A = 5$.