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Mat. Sb., 2019, Volume 210, Number 3, Pages 75–130 (Mi msb9018)  

Is Zaremba's conjecture true?

I. D. Kan

Moscow Aviation Institute (National Research University), Moscow, Russia

Abstract: For finite continued fractions in which all partial quotients lie in the alphabet $\{1,2,3,5\}$, it is shown that the set of denominators not exceeding $N$ has cardinality $\gg N^{0.85}$. A calculation using an analogue of Bourgain-Kontorovich's theorem from 2011 gives $\gg N^{0.80}$.
Bibliography: 25 titles.

Keywords: continued fraction, trigonometric sum, Zaremba's conjecture, partial quotients, continuant, Hausdorff dimension.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-05700-а
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 18-01-00886-а).


DOI: https://doi.org/10.4213/sm9018

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English version:
Sbornik: Mathematics, 2019, 210:3, 364–416

Bibliographic databases:

UDC: 511.36+511.216
MSC: 11А55, 11J70, 11Y65
Received: 16.10.2017 and 29.04.2018

Citation: I. D. Kan, “Is Zaremba's conjecture true?”, Mat. Sb., 210:3 (2019), 75–130; Sb. Math., 210:3 (2019), 364–416

Citation in format AMSBIB
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