### Project Part 07 (G. Larcher): Improved Discrepancy Estimates for Various Classes of Sequences

*(1.SFB funding period)*

It is the main aim of this project to give improved discrepancy estimates for several types of point sequences and point sets in an $s$-dimensional unit-cube. In most cases we will concentrate on the star-discrepancy $D^∗_N$.

In the proposal of this project part we will state 16 concrete research problems in three groups:

**A)** "Metrical'' lower bounds for the discrepancy of good lattice point sets and of digital $(t, m, s)$-nets.

**B)** Discrepancy estimates for hybrid sequences and applications of hybrid sequences.

**C)** Concrete (non-metric) discrepancy estimates.

Concerning estimation of a metrical type or for concrete types of sequences, we especially concentrate ourselves to low-discrepancy sequences which are essential for QMC methods. By working and (at least partly) solving these problems we will give new impulses for various directions of research on discrepancy theory and on low-discrepancy sequences.

*(2.SFB funding period)*

This new application for the second period of the SFB contains projected work partly on problems which remain open after the first period, and partly on new problems related to the new developments and new directions of research initiated in the first period of the SFB. It remains the main aim of this project to give improved discrepancy estimates for several types of point sequences and point sets in the $d$-dimensional unit-cube.

We will state 16 concrete research problems in the following three thematic groups:

- lower bounds for discrepancy

- distribution properties of sequences of the form $(\{ a_n \alpha \})_{n \ge 1}$

- particle motions and applications of hybrid sequences in finance

More so than in the application for the first period, this proposal will contain "high-risk-projects''.