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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 2, Pages 87–144 (Mi izv8032)  

This article is cited in 7 scientific papers (total in 7 papers)

A strengthening of a theorem of Bourgain and Kontorovich

I. D. Kana, D. A. Frolenkovb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: We prove that the set of positive integers contains a positive proportion of numbers satisfying Zaremba's conjecture with $A=7$. This result strengthens a similar theorem of Bourgain and Kontorovich obtained for $A=50$.

Keywords: continued fraction, continuant, exponential sums.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00681-а
11-01-00759-а
12-01-31165
Dynasty Foundation


DOI: https://doi.org/10.4213/im8032

Full text: PDF file (985 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2014, 78:2, 293–353

Bibliographic databases:

Document Type: Article
UDC: 511.321+511.31
MSC: 11A55, 11B05, 11P55
Received: 10.07.2012
Revised: 24.05.2013

Citation: I. D. Kan, D. A. Frolenkov, “A strengthening of a theorem of Bourgain and Kontorovich”, Izv. RAN. Ser. Mat., 78:2 (2014), 87–144; Izv. Math., 78:2 (2014), 293–353

Citation in format AMSBIB
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  • https://doi.org/10.4213/im8032
  • http://mi.mathnet.ru/eng/izv/v78/i2/p87

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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. I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. III”, Izv. Math., 79:2 (2015), 288–310  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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. N. Zh. Nauryzbaev, A. A. Shomanova, N. Temirgaliyev, “On some special effects in theory on numerical integration and functions recovery”, Russian Math. (Iz. VUZ), 62:3 (2018), 84–88  mathnet  crossref  isi
    6. I. D. Kan, “Linear Congruences in Continued Fractions on Finite Alphabets”, Math. Notes, 103:6 (2018), 911–918  mathnet  crossref  crossref  isi  elib
    7. I. D. Kan, “Is Zaremba's conjecture true?”, Sb. Math., 210:3 (2019), 364–416  mathnet  crossref  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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