A Gauss-Kuzmin Theorem for Continued Fractions Associated with Nonpositive Integer Powers of an Integer<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math>

Lascu, Dan

doi:10.1155/2014/984650

Cited by 6 publications

(3 citation statements)

References 20 publications

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“…A proof of this distribution and of the type I Khinchine constant for each integer base b, using ergodic theory, can be found in [6]. Additionally, it is likely that the proofs in Appendices A and B for the type III continued logarithm distribution and logarithmic Khinchine constant could be appropriately adjusted to prove these results.…”

Section: Experimentally Determining the Type I Distributionmentioning

confidence: 99%

Continued Logarithms and Associated Continued Fractions

Borwein

Calkin

Lindstrom

et al. 2016

Experimental Mathematics

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We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base b. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchine's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchine constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize past continued logarithms.

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Section: Experimentally Determining the Type I Distributionmentioning

confidence: 99%

Continued Logarithms and Associated Continued Fractions

Borwein

Calkin

Lindstrom

et al. 2016

Experimental Mathematics

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“…Apart from the regular continued fraction expansion, very many other continued fraction expansions were studied [14,16]. By such a development, generalizations of these problems for non-regular continued fractions are also called as the Gauss-Kuzmin problem and the Gauss-Kuzmin-Lévy problem [8,10,11,17,18,19,20].…”

Section: Gauss' Problem and Its Progressmentioning

confidence: 99%

Ergodic Properties Of $θ$-Expansions And A Gauss-Kuzmin-Type Problem

Lascu¹,

Nicolae²,

Batran³

2014

Preprint

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“…In this paper we consider a non-regular continued fraction expansions introduced by Chan [1]. In fact, Chan considered some continued fraction expansions related to random Fibonacci-type sequences which were studied in detail by Sebe and Lascu in [6,4,5].…”

Section: Introductionmentioning

confidence: 99%

New Perspectives in the Metric Theory of Continued Fraction Expansion Related to Fibonacci Type Sequences

Lascu¹

2016

SBNA

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A survey of the metric theory of the continued fraction expansions related to random Fibonacci Type sequences discussed by Sebe and Lascu is given. The limit properties of these expansions have been studied. A Wirsing-type approach to the Perron-Frobenius operator of the generalized Gauss map under its invariant measure allows us to get close to the optimal convergence rate. Actually, we obtain upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-Lévy problem for these expansions.

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A Gauss-Kuzmin Theorem for Continued Fractions Associated with Nonpositive Integer Powers of an Integerm≥2

Abstract: We consider a family {τ m : m ≥ 2} of interval maps which are generalizations of the Gauss transformation. For the continued fraction expansion arising from τ m, we solve a Gauss-Kuzmin-type problem.

Cited by 6 publications

References 20 publications

Continued Logarithms and Associated Continued Fractions

Continued Logarithms and Associated Continued Fractions

Ergodic Properties Of $θ$-Expansions And A Gauss-Kuzmin-Type Problem

New Perspectives in the Metric Theory of Continued Fraction Expansion Related to Fibonacci Type Sequences

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