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On the uniform distribution of certain sequences and Benford’s law. (English) Zbl 0649.10044
The author proves: If \(v_ n=x_ n+g(y_ n)\) is a sequence of real numbers, where g(t) is a real function with period 1 and the two- dimensional sequence \((x_ 1,y_ 1)\), \((x_ 2,y_ 2),..\). is uniformly distributed mod 1, then \(v_ n\) is uniformly distributed mod 1. This result is of consequence for linear recurrences. Let \(u_ n\) be a second order linear recursive sequence defined by \(u_{n+2}=au_{n+1}+bu_ n\) (n\(\geq 0)\), where a, b, \(u_ 0\), \(u_ 1\) are given real numbers with a \(2+4b<0\). Then, under some assumptions, the sequence \(\log_{10}| u_ n|\) is uniformly distributed mod 1. In the easier case a \(2+4b>0\) this was proved earlier by several authors.
Reviewer: P.Kiss

MSC:
11K06 General theory of distribution modulo \(1\)
11B37 Recurrences
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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References:
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