On the problem of periodicity of continued fraction expansions of [IMG align=ABSMIDDLE alt=$ \sqrt{f}$]tex_sm_4994_img1[/IMG] for cubic polynomials over number fields

2022 ◽  
Vol 213 (3) ◽  
Author(s):  
Vladimir Petrovich Platonov ◽  
Vladimir Sergeevich Zhgoon ◽  
Maksim Maksimovich Petrunin
Keyword(s):  
Continued Fraction ◽  
Number Fields ◽  
Cubic Polynomials ◽  
Continued Fraction Expansions

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

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

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.

Sign in / Sign up

Export Citation Format

Share Document