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


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