On an important family of inequalities of Niederreiter involving exponential sums

The inequality of Erdos-Turan-Koksma is a fundamental tool to bound the discrepancy of a sequence in the s-dimensional unit cube [0, 1), s ≥ 1, in terms of exponential sums. In an impressive series of papers, Harald Niederreiter has established variants of this inequality and has proved bounds for the discrepancy for various sequences and point sets, in the context of pseudo-random number generation and in quasi-Monte Carlo methods. These results have been an important breakthrough, because they marked the starting point of a thorough theoretical correlation analysis of pseudo-random numbers. Niederreiter’s technique also prepared for the study of digital sequences, which are central to modern quasi-Monte Carlo methods. In this contribution, we present an overview of these concepts and prove a hybrid version of the inequality of Erdos-Turan-Koksma, thereby extending a recent result of Niederreiter.