A reinforcement of the Bourgain-Kontorovich's theorem

Dmitriy Frolenkov; Igor D. Kan

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arXiv · 1207.5168

A reinforcement of the Bourgain-Kontorovich's theorem

Dmitriy Frolenkov·Igor D. Kan

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Abstract

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$ 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 $\N.$ In this paper the same theorem is proved with A=7.

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Dmitriy Frolenkov, Igor D. Kan. 2012-07-21. A reinforcement of the Bourgain-Kontorovich's theorem. https://arxiv.org/abs/1207.5168

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