Discrepancy of digital nets and sequences (Subprojekt des NFN S-96)

This project is devoted to the analysis of various notions of discrepancy of digital nets and sequences over a finite field. Digital nets were introduced by Niederreiter and at the moment they provide the most efficient method to generate point sets with small discrepancy. Today such point sets are most frequently used for quasi-Monte Carlo (QMC) quadrature rules, which are successfully used in many applications in financial mathematics, physics and engineering. Firstly we will focus on the classical notions of discrepancy, as for example the star discrepancy or the L2 discrepancy. For example, we want to extend the method of Walsh series, where estimates for digital (0,m,2)-nets in base 2 were considerably improved in the context of symmetrised digital sequences or digital shift-nets. Secondly we will will deal with a new notion of discrepancy, the so-called weighted discrepancy. Our aim is to give good and effectively useful bounds for the weighted discrepancy of digital nets and to construct digital nets with "small" weighted discrepancy. Further we are interested in the worst-case error for integration with QMC rules using digital point sets as nodes.