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.
