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

Author

Listed:
  • Dan Lascu

    (Mircea cel Batran Naval Academy, 1 Fulgerului, 900218 Constanta, Romania
    These authors contributed equally to this work.)

  • Gabriela Ileana Sebe

    (Faculty of Applied Sciences, Politehnica University of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania
    Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Sept. 13, 050711 Bucharest, Romania
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Dan Lascu & Gabriela Ileana Sebe, 2021. "A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms," Mathematics, MDPI, vol. 9(3), pages 1-14, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:255-:d:488295
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