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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)
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 $.
Received: April 16, 2016
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
Linking options:
https://www.mathnet.ru/eng/tm3765https://doi.org/10.1134/S0371968517010101 https://www.mathnet.ru/eng/tm/v296/p133
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Abstract page: | 305 | Full-text PDF : | 44 | References: | 55 | First page: | 13 |
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