Open Access
VOL. 48 | 2006 Entropy quotients and correct digits in number-theoretic expansions
Wieb Bosma, Karma Dajani, Cor Kraaikamp

Editor(s) Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy

IMS Lecture Notes Monogr. Ser., 2006: 176-188 (2006) DOI: 10.1214/074921706000000202


Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of approximation of decimal and regular continued fraction expansions (almost everywhere) to the quotient of the entropies of their dynamical systems. He used detailed knowledge of the continued fraction operator. In 2001, a generalization of Lochs’ result was given by Dajani and Fieldsteel, Equipartition of interval partitions and an application to number theory, describing the rate at which the digits of one number-theoretic expansion determine those of another. Their proofs are based on covering arguments and not on the dynamics of specific maps. In this paper we give a dynamical proof for certain classes of transformations, and we describe explicitly the distribution of the number of digits determined when comparing two expansions in integer bases. Finally, using this generalization of Lochs' result, we estimate the unknown entropy of certain number theoretic expansions by comparing the speed of convergence with that of an expansion with known entropy.


Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1128.11039
MathSciNet: MR2306199

Digital Object Identifier: 10.1214/074921706000000202

Primary: 11K55 , 28D20
Secondary: 11K16 , 11K50

Keywords: Continued fraction , digits , Entropy , Expansion , radix

Rights: Copyright © 2006, Institute of Mathematical Statistics

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