## 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:

### 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.