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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Volume 299, Pages 170–191
DOI: https://doi.org/10.1134/S0371968517040112
(Mi tm3825)
 

This article is cited in 1 scientific paper (total in 1 paper)

Haas–Molnar continued fractions and metric Diophantine approximation

Liangang Maa, Radhakrishnan Nairb

a Department of Mathematics, Binzhou University, Huanghe 5 road No. 391, City of Binzhou, Shandong Province, P.R. China
b Department of Mathematical Sciences, The University of Liverpool, Mathematical Sciences Building, Liverpool L69 7ZL, UK
Full-text PDF (307 kB) Citations (1)
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Abstract: Haas–Molnar maps are a family of maps of the unit interval introduced by A. Haas and D. Molnar. They include the regular continued fraction map and A. Renyi's backward continued fraction map as important special cases. As shown by Haas and Molnar, it is possible to extend the theory of metric diophantine approximation, already well developed for the Gauss continued fraction map, to the class of Haas–Molnar maps. In particular, for a real number $x$, if $(p_n/q_n)_{n\geq 1}$ denotes its sequence of regular continued fraction convergents, set $\theta _n(x)=q_n^2|x- p_n/q_n|$, $n=1,2\dots $. The metric behaviour of the Cesàro averages of the sequence $(\theta _n(x))_{n\geq 1}$ has been studied by a number of authors. Haas and Molnar have extended this study to the analogues of the sequence $(\theta _n(x))_{n\geq 1}$ for the Haas–Molnar family of continued fraction expansions. In this paper we extend the study of $(\theta _{k_n}(x))_{n\geq 1}$ for certain sequences $(k_n)_{n\geq 1}$, initiated by the second named author, to Haas–Molnar maps.
Keywords: Haas–Molnar continued fractions, subsequence ergodic theory.
Received: August 4, 2016
English version:
Proceedings of the Steklov Institute of Mathematics, 2017, Volume 299, Pages 157–177
DOI: https://doi.org/10.1134/S0081543817080119
Bibliographic databases:
Document Type: Article
UDC: 511.72
MSC: 11K60, 11J83, 37E30
Language: Russian
Citation: Liangang Ma, Radhakrishnan Nair, “Haas–Molnar continued fractions and metric Diophantine approximation”, Analytic number theory, On the occasion of the 80th anniversary of the birth of Anatolii Alekseevich Karatsuba, Trudy Mat. Inst. Steklova, 299, MAIK Nauka/Interperiodica, Moscow, 2017, 170–191; Proc. Steklov Inst. Math., 299 (2017), 157–177
Citation in format AMSBIB
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\by Liangang~Ma, Radhakrishnan~Nair
\paper Haas--Molnar continued fractions and metric Diophantine approximation
\inbook Analytic number theory
\bookinfo On the occasion of the 80th anniversary of the birth of Anatolii Alekseevich Karatsuba
\serial Trudy Mat. Inst. Steklova
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\vol 299
\pages 170--191
\publ MAIK Nauka/Interperiodica
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  • This publication is cited in the following 1 articles:
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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