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 $\frac{b}{d}=[d_1,d_2,\dots ,d_k],$ 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 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 $\mathcal{A}$ $${\mathfrak{C}}_{\mathcal{A}}=\left\{[d_1,\dots ,d_j,\dots ]:d_j\in \mathcal{A},j=1,\dots \right\}$$ and let $\delta$ be the Hausdorff dimension of ${\mathfrak{C}}_{\mathcal{A}}.$ Naw this result proved with $A=4$ and $\delta >0.7807\dots$.

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