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On the nature of Benford’s law. (English) Zbl 0983.11046
Summary: We study multiplicative and affine sequences of real numbers defined by
\(N(j+1)=\zeta(j)N(j)+\eta(j)\), where \(\{\zeta(j)\}\) and \(\{\eta(j)\}\) are sequences of positive real numbers (in the multiplicative case \(\eta(j)=0\) for all \(j\)). We investigate the conditions under which the leading digits \(k\) of \(\{N(j)\}\) have the following probability distribution, known as Benford’s Law, \(P(k)=\log_{10}((k+1)/k)\). We present two main results. First, we show that contrary to the usual assumption in the literature, \(\{\zeta(j)\}\) does not necessarily need to come from a chaotic or independent random process for Benford’s Law to hold. The multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing. Second, we give conditions under which the distribution of the first digits of an affine process displays Benford’s law. Our proofs use techniques from ergodic theory.

11K31 Special sequences
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Full Text: DOI
[1] Raimi, R., The first digit problem, Am. math. mon., 83, 521-538, (1976) · Zbl 0349.60014
[2] Adhikari, A.K.; Sarkar, B.P., Distribution of most significant digit in certain functions whose arguments are random variables, Sankhyā ser. B, 30, 47-58, (1968)
[3] Adhikari, A.K.; Sarkar, B.P., Some results on the distribution of the most significant digit, Sankhyā ser. B, 31, 413-420, (1969)
[4] Arnold, V.; Avez, A., Ergodic problems of classical mechanics, (1988), Addison-Wesley Amsterdam · Zbl 0715.70004
[5] Keynes, H.B.; Newton, D., Ergodic measures for nonabelian compact group extensions, Compositio math., 32, 53-70, (1976) · Zbl 0318.28006
[6] W. Parry, M. Pollicott, Stability of mixing for toral extensions of hyperbolic systems, Trudy Mat. Inst. Steklov. 216 (1997) 354-363 (Din. Sist. i Smezhnye Vopr.). · Zbl 0988.37034
[7] Field, M.J.; Parry, W., Stable ergodicity of skew extensions by compact Lie groups, Topology, 38, 1, 167-187, (1999) · Zbl 0924.58045
[8] Pietronero, L.; Tosatti, E.; Tossati, V.; Vespignani, A., Explaining the uneven distribution of numbers in nature: the laws of benford and Zipf, Physica A, 293, 297-304, (2001) · Zbl 0978.60032
[9] Nicol, M.; Melbourne, I.; Ashwin, P., Euclidean extensions of dynamical systems, Nonlinearity, 14, 275-300, (2001) · Zbl 1183.37043
[10] Walters, P., Ergodic theory, (1975), Springer Berlin · Zbl 0299.28012
[11] Zipf, G.K., Human behaviour and the principle of least effort, (1949), Addison-Wesley Cambridge
[12] Gutenberg, B.; Richter, C.F., Seismicity of the Earth, (1949), Princeton University Press Princeton, NJ
[13] Manrubia, S.C.; Zanette, D.H., Stochastic multiplicative processes with reset events, Phys. rev. E, 59, 4945-4948, (1999)
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