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Izvestiya: Mathematics, 2016, Volume 80, Issue 6, Pages 1094–1117
DOI: https://doi.org/10.1070/IM8360 (Mi im8360) 		 
This article is cited in 13 scientific papers (total in 13 papers)

A strengthening of a theorem of Bourgain and Kontorovich. IV

I. D. Kan

Moscow Aviation Institute (State University of Aerospace Technologies)
English version PDF (436 kB) Citations (13)
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DOI: https://doi.org/10.1070/IM8360
Abstract: We prove that the denominators of finite continued fractions all of whose partial quotients belong to the alphabet $\{1,2,3,4\}$ form a set of positive density. The analogous theorem was known earlier only for alphabets of larger cardinality. The first result of this kind was obtained in 2011 for the alphabet $\{1,2,\dots,50\}$ by Bourgain and Kontorovich. In 2013, the present author, together with Frolenkov, proved the corresponding theorem for the alphabet $\{1,2,3,4,5\}$. A 2014 result of the present author dealt with the alphabet $\{1,2,3,4,10\}$.
Keywords: continued fraction, continuant, trigonometric sum, Zaremba's conjecture.
Funding agency Grant number
Russian Foundation for Basic Research 15-01-05700-a
The research was supported by RFBR (grant no. 15-01-05700-a).
Received: 23.02.2015
Revised: 22.01.2016
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2016, Volume 80, Issue 6, Pages 103–126
DOI: https://doi.org/10.4213/im8360
Bibliographic databases:
Document Type: Article
UDC: 511.321+511.31
MSC: Primary 11J70; Secondary 11A55, 11L07
Language: English
Original paper language: Russian
Citation: I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. IV”, Izv. RAN. Ser. Mat., 80:6 (2016), 103–126; Izv. Math., 80:6 (2016), 1094–1117
Citation in format AMSBIB
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    This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Russian version PDF:63
    English version PDF:29
    References:57
    First page:26
     
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