A reinforcement of the Bourgain-Kontorovich's theorem

2012/07/21 by Dmitriy Frolenkov, Frolenkov, Dmitriy, Igor D. Kan +1
Mathematics · #11J70 #11L03 #11P55 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11J70 #msc:11L03 #msc:11P55

paper · pdf · doi:10.48550/arxiv.1207.5168

69 pages, 1 figure

arxiv created 2012/07/21 · arxiv updated 2012/07/24

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 the same theorem is proved with A=7.

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