  Zaremba's conjecture (1971) states that every positive integer number $d$ can
be represented as a denominator (continuant) of a finite continued fraction
$\frac{b}{d}=[d_1,d_2,\ldots,d_{k}],$ with all partial quotients
$d_1,d_2,\ldots,d_{k}$ 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
2014 Kan and Frolenkov proved this result with $A=5.$ Let
$\mathfrak{C}_{\mathcal{A}}$ be the set of infinite continued fractions whose
partial quotients belong to $\mathcal{A}$
$$\mathfrak{C}_{\mathcal{A}}=\left\{[d_1,\ldots,d_j,\ldots]:
d_j\in\mathcal{A},\,j=1,\ldots\right\}$$ and let $\delta$ be the Hausdorff
dimension of $\mathfrak{C}_{\mathcal{A}}.$ Naw this result proved with $A=4$
and $\delta>0.7807\ldots$.
