I. D. Kan, “Is Zaremba's conjecture true?”, Sb. Math., 210:3 (2019), 364

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Sbornik: Mathematics, 2019, Volume 210, Issue 3, Pages 364–416
DOI: https://doi.org/10.1070/SM9018 (Mi sm9018)

This article is cited in 8 scientific papers (total in 8 papers)

Is Zaremba's conjecture true?

I. D. Kan

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

English version PDF (1076 kB) Citations (8) Russian version article

References:

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DOI: https://doi.org/10.1070/SM9018

Abstract: For finite continued fractions in which all partial quotients lie in the alphabet

{1, 2, 3, 5}

$\{1,2,3,5\}$ , it is shown that the set of denominators not exceeding

N

$N$ has cardinality

≫ N^{0.85}

$\gg N^{0.85}$ . A calculation using an analogue of Bourgain-Kontorovich's theorem from 2011 gives

≫ N^{0.80}

$\gg N^{0.80}$ .
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).

Received: 16.10.2017 and 29.04.2018

Bibliographic databases:

Document Type: Article

UDC: 511.36+511.216

MSC: 11А55, 11J70, 11Y65

Language: English

Original paper language: Russian

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

Citation in format AMSBIB

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\paper Is Zaremba's conjecture true?

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\pages 364--416

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Linking options:

https://www.mathnet.ru/eng/sm9018

https://doi.org/10.1070/SM9018

https://www.mathnet.ru/eng/sm/v210/i3/p75

This publication is cited in the following 8 articles:

I. D. Kan, G. Kh. Solov'ev, “System of Inequalities in Continued Fractions from Finite Alphabets”, Math. Notes, 113:2 (2023), 212–219
I. D. Kan, “Modular Generalization of the Bourgain–Kontorovich Theorem”, Math. Notes, 114:5 (2023), 785–796
I. D. Kan, “Strengthening of the Burgein–Kontorovich theorem on small values of Hausdorff dimension”, Funct. Anal. Appl., 56:1 (2022), 48–60
I. D. Kan, V. A. Odnorob, “Linear Inhomogeneous Congruences in Continued Fractions on Finite Alphabets”, Math. Notes, 112:3 (2022), 424–435
M. Pollicott, P. Vytnova, “Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups”, Trans. Amer. Math. Soc. Ser. B, 9:35 (2022), 1102
I. D. Kan, “A strengthening of the Bourgain-Kontorovich method: three new theorems”, Sb. Math., 212:7 (2021), 921–964
I. D. Kan, V. A. Odnorob, “Inversions of Hölder's Inequality”, Math. Notes, 110:5 (2021), 700–708
I. D. Kan, “Usilenie odnoi teoremy Burgeina – Kontorovicha”, Dalnevost. matem. zhurn., 20:2 (2020), 164–190

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Statistics & downloads:
Abstract page:	686
Russian version PDF:	127
English version PDF:	43
References:	61
First page:	44

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