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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Volume 296, Pages 133–139
DOI: https://doi.org/10.1134/S0371968517010101
(Mi tm3765)
 

This article is cited in 10 scientific papers (total in 10 papers)

A strengthening of a theorem of Bourgain and Kontorovich. V

I. D. Kan

Moscow Aviation Institute (National Research University)
References:
Abstract: It is proved that the denominators of finite continued fractions all of whose partial quotients belong to an arbitrary finite alphabet $\mathcal A$ with parameter $\delta >0.7807\dots $ (i.e., such that the set of infinite continued fractions with partial quotients from this alphabet is of Hausdorff dimension $\delta $ with $\delta >0.7807\dots $) contain a positive proportion of positive integers. Earlier, a similar theorem has been obtained only for alphabets with somewhat greater values of $\delta $. Namely, the first result of this kind for an arbitrary finite alphabet with $\delta >0.9839\dots $ is due to Bourgain and Kontorovich (2011). Then, in 2013, D.A. Frolenkov and the present author proved such a theorem for an arbitrary finite alphabet with $\delta >0.8333\dots $. The preceding result of 2015 of the present author concerned an arbitrary finite alphabet with $\delta >0.7862\dots $.
Funding agency Grant number
Russian Foundation for Basic Research 15-01-05700-a
This work was supported by the Russian Foundation for Basic Research, project no. 15-01-05700-a.
Received: April 16, 2016
English version:
Proceedings of the Steklov Institute of Mathematics, 2017, Volume 296, Pages 125–131
DOI: https://doi.org/10.1134/S0081543817010102
Bibliographic databases:
Document Type: Article
UDC: 511.321+511.31
Language: Russian
Citation: I. D. Kan, “A strengthening of a theorem of Bourgain and Kontorovich. V”, Analytic and combinatorial number theory, Collected papers. On the occasion of the 125th anniversary of the birth of Academician Ivan Matveevich Vinogradov, Trudy Mat. Inst. Steklova, 296, MAIK Nauka/Interperiodica, Moscow, 2017, 133–139; Proc. Steklov Inst. Math., 296 (2017), 125–131
Citation in format AMSBIB
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\paper A strengthening of a theorem of Bourgain and Kontorovich. V
\inbook Analytic and combinatorial number theory
\bookinfo Collected papers. On the occasion of the 125th anniversary of the birth of Academician Ivan Matveevich Vinogradov
\serial Trudy Mat. Inst. Steklova
\yr 2017
\vol 296
\pages 133--139
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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  • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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