Vol. 104, No. 1, 1983

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A general version of van der Corput’s difference theorem

Rudolf J. Taschner

Vol. 104 (1983), No. 1, 231–239

Let ω(n) be a real-valued sequence, and let us assume that for all positive integers g the difference-sequences Δgω(n) = ω(n + g) ω(n) are uniformly distributed modulo 1, then ω(n) itself is uniformly distributed modulo 1. This is van der Corput’s difference theorem or the so-called main theorem in the theory of uniform distribution. In this paper I present a rather generalized version of this theorem which not only enables us to prove the original van der Corput theorem, but also the general approximation theorem of Kronecker in its discrete and in its continuous version. Moreover, a few other examples of uniformly distributed sequences can be constructed by this general difference theorem.

Mathematical Subject Classification
Primary: 10K05, 10K05
Received: 2 September 1979
Published: 1 January 1983
Rudolf J. Taschner