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Benford’s law and distribution functions of sequences in $$(0,1)$$. (English) Zbl 1242.11055
Math. Notes 88, No. 4, 449-463 (2010); reprinted from Mat. Zametki 88, No. 4, 485-501 (2010).
In this article the authors find a functional equation for distribution functions of a sequence $$x_n$$ that satisfies the strong Benford law. Let $$G(x_n)$$ be the set of all distribution functions of $$x_n$$. Assume that every $$g(x)\in G(x_n)$$ is continuous at $$x=0$$. Then the authors prove that the sequence $$x_n$$ satisfies the strong Benford law in the base $$b$$ if and only if for every $$g(x)\in G(x_n)$$, $$x=\sum_{i=0}^\infty (g(1/b^{i})-g(1/b^{i+x}))$$ for $$x\in [0,1]$$. Examples of distribution functions of sequences satisfying the functional equation are given. Furthermore, the authors transform some theorems from uniform distribution theory to the language of Benford’s law.
##### MSC:
 11K06 General theory of distribution modulo $$1$$ 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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