A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Dan Lascu and Gabriela Ileana Sebe
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Dan Lascu: Mircea cel Batran Naval Academy, 1 Fulgerului, 900218 Constanta, Romania
Gabriela Ileana Sebe: Faculty of Applied Sciences, Politehnica University of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania

Mathematics, 2021, vol. 9, issue 3, 1-14

Abstract: We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ -expansions, N -continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

Keywords: continued fractions; entropy; measure preserving transformation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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