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A discrepancy problem with applications to linear recurrences. I, II. (English) Zbl 0692.10041

The following theorem is proved in the paper: “Let \(y_ 1,...,y_ s\) be a multiplicative independent system of unimodular complex algebraic numbers and let \(\theta_ k\) be real numbers defined by \(y_ k=e^{2\pi i \vartheta_ k}\) \((k=1,...,s)\). Set \(\vartheta =(\vartheta_ 1,...,\vartheta_ s)\) and let \(\omega =(\omega_ 1,...,\omega_ s)\) be an arbitrary s-tuple of real numbers. Then the discrepancy of the s-dimensional sequence \((x_ n)=(n\vartheta +\omega)\) satisfies the estimate \(D_ N(x_ n)\leq N^{-\delta}\) for any sufficiently large N, where \(\delta\) \((>0)\) depends only on the system \(y_ 1,...,y_ s.''\) This result can be used to answer an approximation problem concerning second order linear recurrences.
Reviewer: P.Kiss

MSC:

11K06 General theory of distribution modulo \(1\)
11B37 Recurrences
11J04 Homogeneous approximation to one number
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References:

[1] A. Baker: The theory of linear forms in logarithms, transcendence theory. Advances and Applications (eds. A. Baker and D. W. Masser), London-New York, Academic Press, 1-27 (1977). · Zbl 0361.10028
[2] P. Kiss: Zero terms in second order linear recurrences. Math. Sem. Notes (Kobe Univ.), 7, 145-152 (1979). · Zbl 0417.10009
[3] P. Kiss: A diophantine approximative property of the second order linear recurrences. Period. Math. Hungar., 11, 281-287 (1980). · Zbl 0458.10011
[4] P. Kiss and R. F. Tichy: Distribution of the ratios of the terms of a second order linear recurrence. Proc. of the Konink. Nederlandse Akad. Weten., ser. A, 89, 79-86 (1986). · Zbl 0586.10006
[5] P. Kiss and Z. Sinka: On the ratios of the terms of second order linear recurrences (to appear). · Zbl 0743.11009
[6] L. Kuipers and H. Niederreiter: Uniform Distribution of Sequences. John Wiley & Sons, New York (1974). · Zbl 0281.10001
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