Entropy quotients and correct digits in number-theoretic expansions.

*(English)*Zbl 1128.11039
Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 176-188 (2006).

Given two different expansions for real numbers, one of the ways in which they may be compared is to ask how many digits of one are needed to determine digits of the other. This may be thought of geometrically, with the first \(n\) digits in one expansion determining some subset of the reals, often an interval, to be compared with the set determined by the first \(m\) digits in the other expansion. This may also be viewed as a coding problem, in which view it is natural to expect that the entropy of the processes underlying the two expansions will determine the asymptotic behaviour.

G. Lochs [Abh. Math. Semin. Univ. Hamb. 27, 142–144 (1964; Zbl 0124.28003)] showed that if \(m(n)\) denotes the number of continued fraction digits determined by the first \(n\) digits in the decimal expansion of a real number, then \({m(n)}/{n}\rightarrow (6\log2\log10)/\pi^2\), the ratio of the entropies. This was generalized by K. Dajani and A. Fieldsteel [Proc. Am. Math. Soc. 129, No. 12, 3453–3460 (2001; Zbl 0999.11041)], who showed a similar result for any two sequences of partitions each obeying the asymptotic equipartition (or Shannon-McMillan-Breiman) theorem by working directly with the measure-theoretic properties of the partitions.

Here a dynamical approach is taken, and the distribution of \(m(n)\) is computed explicitly for expansions in integer bases. A numerical investigation is also made of the entropy of some number-theoretical expansions of unknown entropy by making this comparison with expansions of known entropy.

For the entire collection see [Zbl 1113.60008].

G. Lochs [Abh. Math. Semin. Univ. Hamb. 27, 142–144 (1964; Zbl 0124.28003)] showed that if \(m(n)\) denotes the number of continued fraction digits determined by the first \(n\) digits in the decimal expansion of a real number, then \({m(n)}/{n}\rightarrow (6\log2\log10)/\pi^2\), the ratio of the entropies. This was generalized by K. Dajani and A. Fieldsteel [Proc. Am. Math. Soc. 129, No. 12, 3453–3460 (2001; Zbl 0999.11041)], who showed a similar result for any two sequences of partitions each obeying the asymptotic equipartition (or Shannon-McMillan-Breiman) theorem by working directly with the measure-theoretic properties of the partitions.

Here a dynamical approach is taken, and the distribution of \(m(n)\) is computed explicitly for expansions in integer bases. A numerical investigation is also made of the entropy of some number-theoretical expansions of unknown entropy by making this comparison with expansions of known entropy.

For the entire collection see [Zbl 1113.60008].

Reviewer: Thomas Ward (Norwich)