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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.
11K06 General theory of distribution modulo \(1\)
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Full Text: DOI
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