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

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.

Approximation of functions of several variables by multidimensional $S$-fractions with independent variables

“…The calculation of mathematical constants has been a topic of investigation for mathematicians throughout the centuries [15,Chapter 10]. Some modern applied problems of approximation of numbers by continued fraction expansions are considered in [27,28,31,36]. Here we show how branched continued fractions can be relevant in connection with some of the important mathematical constants.…”

Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions

“…One of the fundamental problems in approaches to finding solutions of such equations is the reconstruction of functions of one or several variables, as well as problems that arise in the development and implementation of effective methods and algorithms for representing and approximating the functions of one or several variables. There are many various tools for representing and approximating the above-mentioned functions, among which, perhaps, one of the most effective are continued fractions [8][9][10][11][12][13][14][15][16][17], and their multidimensional generalizations-branched continued fractions [18][19][20][21][22][23][24][25][26][27][28].…”