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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 2, Pages 77–100 (Mi izv8253)  

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

A strengthening of a theorem of Bourgain and Kontorovich. III

I. D. Kan

Moscow Aviation Institute (State University of Aerospace Technologies)

Abstract: We prove that the set of positive integers contains a positive proportion of denominators of the finite continued fractions all of whose partial quotients belong to the alphabet $\{1,2,3,4,10\}$. The corresponding theorem was previousy known only for the alphabet $\{1,2,3,4,5\}$ and for alphabets of larger cardinality.

Keywords: continued fraction, continuant, trigonometric sum, Zaremba's conjecture.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00681-a
The research was financially supported by RFBR (grant no. 12-01-00681-a).


DOI: https://doi.org/10.4213/im8253

Full text: PDF file (655 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2015, 79:2, 288–310

Bibliographic databases:

UDC: 511.321+511.31
MSC: Primary 11J70; Secondary 11A55, 11L07
Received: 16.05.2014

Citation: I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. III”, Izv. RAN. Ser. Mat., 79:2 (2015), 77–100; Izv. Math., 79:2 (2015), 288–310

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  • https://doi.org/10.4213/im8253
  • http://mi.mathnet.ru/eng/izv/v79/i2/p77

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    Citing articles on Google Scholar: Russian citations, English citations
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    Cycle of papers

    This publication is cited in the following articles:
    1. Moshchevitin N. Murphy B. Shkredov I., “Popular Products and Continued Fractions”, Isr. J. Math.  crossref  mathscinet  isi
    2. I. D. Kan, “Inversion of the Cauchy–Bunyakovskii–Schwarz Inequality”, Math. Notes, 99:3 (2016), 378–381  mathnet  crossref  crossref  mathscinet  isi  elib
    3. I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. IV”, Izv. Math., 80:6 (2016), 1094–1117  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. V”, Proc. Steklov Inst. Math., 296 (2017), 125–131  mathnet  crossref  crossref  mathscinet  isi  elib
    5. I. D. Kan, “Is Zaremba's conjecture true?”, Sb. Math., 210:3 (2019), 364–416  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. I. D. Kan, “Usilenie odnoi teoremy Burgeina – Kontorovicha”, Dalnevost. matem. zhurn., 20:2 (2020), 164–190  mathnet  crossref
    7. I. D. Kan, “A strengthening of the Bourgain-Kontorovich method: three new theorems”, Sb. Math., 212:7 (2021), 921–964  mathnet  crossref  crossref  isi
    8. I. D. Kan, V. A. Odnorob, “Inversions of Hölder's Inequality”, Math. Notes, 110:5 (2021), 700–708  mathnet  crossref  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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