Benford’s law, recurrence relations, and uniformly distributed sequences. II. (Loi de Benford, relations de récurrence et suites équidistribuées. II.) (French) Zbl 1242.11056

Jolissaint, Paul

Elem. Math. 64, No. 1, 21-36 (2009).

This article deals with Benford’s law and uniformly distributed sequences. Let \(b\geq 2\) be an integer. Each real \(x>0\) can be written in one way \(x=M_b(x).b^{e_b(x)}\) where \(e_b(x)\in\mathbb Z\) and \(M_b(x)\) is the mantissa of \(x\) in basis \(b\). Let \((a_n)_{n\geq 1}\subset (0,+\infty)\). The sequence \((a_n)_{n\geq 1}\) is said satisfy Bendford’s law in basis \(b\) if for all \(t\in[1,b)\) if \[ \lim_{N\rightarrow\infty} \frac{|\{1\leq n\leq N\;:\;M_b(a_n)<t\}|}{N}=\log_b(t)\text{\;for all\;}t\in [1,b). \] The sequence \((a_n)_{n\geq 1}\) is said satisfy strong Benford’s law if it satisfy Bendford’s law in every basis \(b\geq 2\). The main result of the author generalizing his previous article [Elem. Math. 60, No. 1, 10–18 (2005; Zbl 1084.11005)] is:
Theorem. Let \(\alpha>0, \xi>0\) and \(\mu\) be real numbers and \(Q\) a function defined on \([1,\infty)\) satisfying:
1) There exists an integer \(k\geq 1\) and a real number \(x_0\geq 1\) such that \(Q\) be \(k\)-times differentiable on \((x_0,+\infty)\);
2) \(\lim_{x\rightarrow\infty} Q^{(k)}\) exists and is a nonzero rational number.
Let \((a_n)_{n\geq 1}\in (0,+\infty)\) be a sequence such that \(\lim_{n\rightarrow\infty}\frac{a_n}{n^\mu\xi^{Q(n)}}=\alpha\).
Then, for every integer \(b\geq 2\) such that \(\log_b(\xi)\in \mathbb R\backslash \mathbb Q\), the sequence \((a_n)_{n\geq 1}\) satisfies Benford’s law in basis \(b\). Moreover, if, for every positive integer \(m\), \(\xi^m\) is not integer, then \((a_n)_{n\geq 1}\) satisfies strong Benford’s law.
The proof relies on the theory of uniformly distributed sequences. On the historic background of Benford’s law see also [A. Berger, L. A. Bunimovich and T. P. Hill, Trans. Am. Math. Soc. 357, No. 1, 197–219 (2005; Zbl 1123.37006)]; [A. Berger and T. P. Hill, “Newton’s Method obeys Bendford’s law”, Am. Math. Mon. 114, No. 7, 588–601 (2007; Zbl 1136.65048)]; [J. P. Delahaye, “L’étonnante loi de Benford”, Pour la Science 351, 90–95 (2007)]; [N. Hungerbühler, “Benfords Gesetz über führende Ziffern”, EducETH, March (2007), http://www.educ.ethz.ch].

Reviewer: Roland Quême (Brax)

MSC:

11K36	Well-distributed sequences and other variations
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
60E99	Distribution theory

Keywords:

Benford’s law; uniformly distributed sequences

Citations:

Zbl 1084.11005; Zbl 1123.37006; Zbl 1136.65048