zbMATH — the first resource for mathematics

On Benford’s law for continued fractions. (English) Zbl 0728.11036
A sequence of real numbers \((q_ n)\) is said to obey Benford’s law, if the decimal logarithms (lg \(q_ n)\) are uniformly distributed modulo 1. H. Jager and P. Liardet [Indagationes Math. 50, 181-197 (1988; Zbl 0655.10045)] have proved Benford’s law for the sequence of denominators \(q_ n(\omega)\) of the continued fraction expansion of a quadratic irrational \(\omega\). In the present article this statement is generalized to regular Hurwitz continued fractions. Furthermore Benford’s law is obtained for almost all real numbers \(\omega\).
Reviewer: R.F.Tichy (Graz)
11K06 General theory of distribution modulo \(1\)
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Full Text: DOI
[1] Herrmann, Asymptotische Gleichverteilungseigenschaften von Summen schwach abhängiger Zufallsgrößen, Math. Nachr. 114 pp 263– (1983) · Zbl 0551.60008 · doi:10.1002/mana.19831140120
[2] Jager, Distributions arithmétiques des dénominators de convergents de fractions continues, Indag. Math. 91 pp 181– (1988) · doi:10.1016/S1385-7258(88)80026-X
[3] Kanemitsu, Proc. of the 5-th Japan-USSR Symposium on Prob. Th. (Lecture notes in Mathematics 1299 pp 158– (1988)
[4] A. Khintchine 1956
[5] Kuipers, Uniform distribution of sequences (1974) · Zbl 0281.10001
[6] O. Perron 1954
[7] Schatte, Zur Verteilung der Mantisse in der Gleitkommadarstellung einer Zufallsgröß, Zeituchr. f. Angew. Math. u. Mech. 58 pp 553– (1973) · Zbl 0267.60025 · doi:10.1002/zamm.19730530807
[8] Schatte, On sums modulo 2\(\deg\) of independent random variables, Math. Nachr. 110 pp 243– (1983) · Zbl 0523.60016 · doi:10.1002/mana.19831100118
[9] Schatte, On mantissa distributions in computing and Benford’s law, J. Inf. Process. Cybern. EIK 24 pp 443– (1988) · Zbl 0662.65040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.