## 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].

### 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
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