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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
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
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
Linking options:
https://www.mathnet.ru/eng/tm3825https://doi.org/10.1134/S0371968517040112 https://www.mathnet.ru/eng/tm/v299/p170
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