s(N) - Uniform Distribution Modulo 1 Original Research Article

In this paper we study the notion of s(N)-uniform distribution of sequences modulo 1 which sharpens resp. quantifies the notion of complete uniform distribution. A trivial necessary condition for the existence of s(N)-u.d. sequences is s(N) = o(N). On the other hand s(N) = o([formula]) implies that almost all sequences are s(N)-u.d. (It seems to be very difficult to fill the gap between those two results.) By a modification of the metric methods it is shown how to construct an s(N)-u.d. sequence if s(N) = o([formula]). This construction is rather involved. So a final section discusses some possibilities of simpler approaches for constructing s(N)-u.d. sequences.