Distribution of sums-of-digits and digital sequences

In the last three years the applicant (G. Larcher) and the main designated project assistant (R. Hofer) have introduced the concept of Niederreiter-Halton sequences (NH sequences). These sequences are generated by combining digital (T,w_i)-sequences in the sense of Niederreiter in different bases to a single point-sequence in the s-dimensional unit-cube. The classical Halton sequences are basic examples of NH sequences. In a series of papers we have investigated this new class of sequences in the s-dimensional unit-cube with respect to their distribution properties. Especially it was tried to study which of the NH sequences are low-discrepancy point-sequences. It turned out that these investigations lead to interesting and basic questions on (weighted) sums-of-digits in different bases. Especially (joint) distribution properties of (weighted) sums-of-digits (in different bases) play an essential role for the investigation of NH sequences. Various works in this direction already also have been carried out by the author and the co-author of this description. It is the principal aim of this project to essentially extend this work, in some sense to complete the study of NH sequences and thereby to answer some challenging questions concerning the distribution of sums-of-digits. The projected work was classified into nine groups of problems. The main topics are the following: Generalization of results of Newman, Drmota and Skalba, Drmota and Stoll on the distribution of sums-of-digits of numbers in arithmetic progressions to arbitrary bases and to weighted sums-of-digits, discrepancy of different types of NH sequences, lower bounds for joint distribution properties of sums-of-digits in different bases (in the sense of Kim), (improved) discrepancy estimates for "hybrid sequences" in the sense of Niederreiter, lower bounds for the discrepancy of the Halton sequences.

High-dimensional, very well distributed point sets are used for simulations in various fields of applications like in techniques, economics, physics, medicine, biology, .... The generation of such point sets is a highly complex mathematical problem. Against first intuition, for example, uniform lattices are not suited for simulation tasks. The most efficient generation methods known until now use certain properties of digit representations. In this project we developed various approaches for the generation of such high-dimensional well-distributed point sets and we analysed these approaches. These investigations frequently led to basic problems from the field of number theory, especially concerning sums-of-digits of the representation of numbers in different bases.In the project we published 33 publications in international journals. Altogether nine researchers (four of them female) worked in this project.