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