Convergence rate for R\'enyi-type continued fraction expansions
Gabriela Ileana Sebe, Dan Lascu
TL;DR
This paper analyzes the convergence rate of Rényi-type continued fraction expansions using a Wirsing-type approach, providing bounds that nearly solve the Gauss-Kuzmin-Lévy problem.
Contribution
It introduces a new method to estimate convergence rates of Rényi-type continued fractions, improving understanding of their optimality and bounds.
Findings
Established upper and lower bounds for convergence rate
Provided near-optimal solutions to the Gauss-Kuzmin-Lévy problem
Enhanced the theoretical understanding of Rényi-type continued fractions
Abstract
This paper continues our investigation of Renyi-type continued fractions studied in \cite{Sebe&Lascu-2018}. A Wirsing-type approach to the Perron-Frobenius operator of the R\'enyi-type continued fraction transformation under its invariant measure allows us to study the optimality of the convergence rate. Actually, we obtain upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-L\'evy problem.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Mathematical Analysis and Transform Methods · Mathematical functions and polynomials