A strengthening of a theorem of Bourgain-Kontorovich-V

TL;DR

This paper discusses recent advances in Zaremba's conjecture, focusing on the density of numbers with bounded partial quotients and improvements in bounds for the Hausdorff dimension of related continued fraction sets.

Contribution

The paper strengthens previous results by improving bounds on the partial quotient size and the Hausdorff dimension for sets related to Zaremba's conjecture.

Findings

Positive proportion of numbers satisfy Zaremba's conjecture with A=50

Result extended to A=5 with Hausdorff dimension > 0.7807

Improved bounds on partial quotient size and dimension

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 $\delta>0.7807\ldots$.