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

2012/10/15 by Dmitriy Frolenkov, Frolenkov, Dmitriy, Игорь Давидович Кан +2
Computer Science · Mathematics · #11J70 #11P55 #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:11J70 #msc:11P55 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1210.4204

13 pages,1 figure

arxiv created 2012/10/15 · openalex publication_date 2012/10/15 · arxiv updated 2012/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator (continuant) of a finite continued fraction (b)/(d)=[d₁,d₂,...,d_k], whose partial quotients d₁,d₂,...,d_k belong to a finite alphabet \A⊆\N. In this paper it is proved for an alphabet \A, such that the Hausdorff dimension δ_\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.

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