A reinforcement of the Bourgain-Kontorovich's theorem by elementary methods II

2013/03/16 by Dmitriy Frolenkov, Frolenkov, Dmitriy, Игорь Давидович Кан +2
Mathematics · #11J70 #11L05 #11P55 #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Number Theory (math.NT) #math.NT #msc:11J70 #msc:11L05 #msc:11P55

paper · pdf · doi:10.48550/arxiv.1303.3968

paper in English, 57 pages

openalex publication_date 2013/03/16 · arxiv created 2013/06/03 · arxiv updated 2013/06/04 · openalex created_date 2024/04/11 · 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], with all partial quotients d₁,d₂,...,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,using only elementary methods, the same theorem is proved with A=5.

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