A strengthening of a theorem of Bourgain-Kontorovich-V

Abstract

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,\ldots,d_{k}],$ 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 $\delta>0.7807\ldots$.