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

2020/05/01 by Dan Lascu, Lascu, Dan, Gabriela Ileana Sebe +1
Computer Science · Mathematics · #11J70 #37A35 #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Numerical Methods and Algorithms

paper · pdf · doi:10.48550/arxiv.2005.00380

openalex publication_date 2020/05/01 · openalex created_date 2021/02/01 · openalex updated_date 2026/07/28

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 aimed 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.

Related

• Lochs-type theorems beyond positive entropy
• Random Lochs' Theorem
• The entropy of alpha-continued fractions: analytical results
• The random continued fraction transformation
• Invariant measures for continued fraction algorithms with finitely many\n digits