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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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