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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.
Full Text: DOI

References:

[1] Brown, Fibonacci Quart. 8 pp 482– (1970)
[2] and , Uniform distribution of sequences, J. Wiley & Sons New York 1974 · Zbl 0281.10001
[3] Kuipers, Fibonacci Quart. 11 pp 292– (1973)
[4] Nagasaka, Ann. Inst. Stat. Math. 36 pp 337– (1984)
[5] Nagasaka, Ann. Inst. Stat. Math.
[6] Schatte, Math. Nachr. 113 pp 237– (1983)
[7] , , 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)
[8] Kiss, Indag. Math. 89 pp 79– (1986) · doi:10.1016/1385-7258(86)90008-9
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