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

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