Manual Metrical Theory of Continued Fractions (Mathematics and Its Applications)

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The random continued fraction transformation - IOPscience

Sets of exact approximation order by rational numbers II. Theory 3 , 9 — Sets of exact approximation order by rational numbers III. Hausdorff dimensions of Cantor sets. Hausdorff dimension of real numbers with bounded digits averages. Springer , New York , Fractal Geometry, Mathematical Foundations and Applications. Wiley , Chichester , The fractional dimensional theory of continued fractions. Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems.

An Introduction to the Theory of Numbers , 5th edn. Clarendon Press , Oxford , A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets. Number Theory 58 , 9 — A problem in the fractional dimension theory of continued fractions. Kluwer Academic Publishers , Dordrecht , Zur metrischen Theorie der diopahantischen Approximationen.

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Diophantische Approximationen und Hausdorffsches Mass. The author declares that there is no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U. Journal List ScientificWorldJournal v. Published online Feb Received Aug 21; Accepted Dec 5. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Introduction Chan considered some continued fraction expansions related to random Fibonacci-type sequences [ 1 , 2 ].

Gauss' Problem and Its Progress One of the first and still one of the most important results in the metrical theory of continued fractions is the so-called Gauss-Kuzmin theorem. Chan's Continued Fraction Expansions In this paper, we consider a generalization of the Gauss transformation and prove an analogous result.

1. Introduction

Known Results and Applications For Chan's continued fraction expansions, we show known results and their applications in this subsection. As it is known, the Fibonacci sequence F n is recursively defined as follows: Main Theorem and Its Consequences 1. Main Theorem We show our main theorems in this subsection. Necessary Lemmas In this subsection, we show some lemmas.

Continued Fractions 1: Introduction and Basic Examples - LearnMathsFree

Proof — Let q m be as in Acknowledgment The author would like to express sincere thanks to the anonymous referees. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. The asymptotic growth rate of random Fibonacci type sequences I. The asymptotic growth rate of random Fibonacci type sequences II.

Springer Series in Computational Mathematics. On a problem of Gauss. On a Gauss-Kuzmin-type problem for a family of continued fraction expansions. Journal of Number Theory. International Journal of Mathematics and Mathematical Sciences. A Wirsing-type approach to some continued fraction expansion. Geometry Balkan Press; Recent advances in the metric theory of a new continued fraction expansion; pp. Iosifescu M, Grigorescu S. Dependence with Complete Connections and Its Applications.

Cambridge University Press; Cambridge Tracts in Mathematics. Iosifescu M, Kraaikamp C. Metrical Theory of Continued Fractions. Iosifescu M, Sebe GI. Mathematical Analysis and Applications. American Institute of Physics; Convergence rate for a continued fraction expansion related to Fibonacci type sequences. Tokyo Journal of Mathematics. Export citation Request permission. Ergodic Theory and Information.

On Khintchine exponents and Lyapunov exponents of continued fractions

Palermo 27 , — Entropy and counting correct digits. Geometric Dynamics Lecture Notes in Mathematics, Springer , Berlin , , pp. Hausdorff dimension of sets arising in number theory. An Introduction to Diophantine Approximation. Cambridge University Press , New York , Springer , New York , Fractal Geometry, Mathematical Foundations and Application. Sur les dimensions de mesures. On the distribution of long-term average on the symbolic space. Recurrence, dimension and entropy. The fractional dimensional theory of continued fractions.


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Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. An Introduction to the Theory of Numbers , 5th edn. Oxford University Press , New York , The Hausdorff dimensions of some continued fraction Cantor sets. Number Theory 33 2 , — A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets. Number Theory 58 1 , 9 — Zur metrischen Theorie der diophantischen Approximationen.

Computing the dimension of dynamically defined sets: E 2 and bounded continued fractions. On the density of Hausdorff dimensions of bounded type continued fraction sets: Geometric barycentres of invariant measures for circle maps. A multifractal analysis for Stern—Brocot intervals, continued fractions and Diophantine growth rates.

University of Chicago Press , Chicago, London ,