A note on the reinforcement of the Bourgain-Kontorovich's theorem

TL;DR

This paper advances the understanding of Zaremba's conjecture by proving that for certain alphabets, a positive proportion of integers can be represented as denominators of continued fractions with partial quotients from that alphabet, improving previous results.

Contribution

It strengthens the Bourgain-Kontorovich theorem by showing a positive density of such denominators for alphabets with specific Hausdorff dimensions.

Findings

Proves positive proportion of integers satisfying Zaremba's conjecture for certain alphabets.

Improves previous reinforcement of Bourgain-Kontorovich's theorem.

Connects Hausdorff dimension of continued fraction sets to number representation density.

Abstract

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ whose partial quotients $d_1,d_2,...,d_{k}$ belong to a finite alphabet $\A\subseteq\N.$ In this paper it is proved for an alphabet $\A,$ such that the Hausdorff dimension $\delta_{\A}$ of the set of infinite continued fractions whose partial quotients belong to $\A,$ that the set of numbers $d,$ satisfying Zaremba's conjecture with the alphabet $\A,$ has positive proportion in $\N.$ The result improves our previous reinforcement of the corresponding Bourgain-Kontorovich's theorem.