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Transactions of the American Mathematical Society

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Almost-arithmetic progressions and uniform distribution

Author: H. Niederreiter
Journal: Trans. Amer. Math. Soc. 161 (1971), 283-292
MSC: Primary 10.33
MathSciNet review: 0284406
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Abstract: In a recent paper, P. E. O'Neil gave a new criterion for uniform distribution modulo one in terms of almost-arithmetic progressions. We investigate the relation between almost-arithmetic progressions and uniformly distributed sequences from a quantitative point of view. An upper bound for the discrepancy of almost-arithmetic progressions is given which is shown to be best possible. Estimates for more general sequences are also obtained. As an application, we prove a quantitative form of Fejér's theorem on the uniform distributivity of slowly increasing sequences.

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Keywords: Almost-arithmetic progressions, uniform distribution modulo one, discrepancy, convex programming
Article copyright: © Copyright 1971 American Mathematical Society

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