On the distribution properties of Niederreiter-Halton sequences (Q999723)

This paper deals with uniform distribution of Niederreiter-Halton sequences. A Niederreiter-Halton sequence is an infinite sequence of points in the \(s\)-dimensional, half-open unit cube, which is obtained by juxtaposing several \((\mathbf{T},s)\)-sequences in the sense of Larcher and Niederreiter. These \((\mathbf{T},s)\)-sequences are usually constructed over a certain integer base \(q\) by methods of linear algebra. The crucial idea in the definition of Niederreiter-Halton sequences is that the component sequences are sequences over different (prime) bases \(q_1,\ldots,q_v\). The author of this paper studies conditions under which it can be shown that Niederreiter-Halton sequences are uniformly distributed modulo one and shows the rather surprising fact that uniform distribution of a Niederreiter-Halton sequence is equivalent to uniform distribution of the component sequences in the different bases. The proof of this result is based on an analysis of weighted sum-of-digits functions of integers.