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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:
  Brown, Fibonacci Quart. 8 pp 482– (1970)  and , Uniform distribution of sequences, J. Wiley & Sons New York 1974 · Zbl 0281.10001  Kuipers, Fibonacci Quart. 11 pp 292– (1973)  Nagasaka, Ann. Inst. Stat. Math. 36 pp 337– (1984)  Nagasaka, Ann. Inst. Stat. Math.  Schatte, Math. Nachr. 113 pp 237– (1983)  , , and , On Benford’s law: The first digit problem, Proceedings of the 5th Japan–USSR Symposium on Probability Theory 1986, Lecture Notes in Mathematics, Springer-Verlag Berlin–New York (to appear)  Kiss, Indag. Math. 89 pp 79– (1986)
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