From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results. On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput's ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter's $(t,s)$-sequences

Explicit constructions of point sets and sequences with low discrepancy

In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of finite point sets in the unit cube of arbitrary dimension. Later various authors extended Roth's result to lower bounds also for the $\mathcal{L}_q$ discrepancy and for infinite sequences. While it was known already from the early 1980s on that Roth's lower bound is best possible in the order of magnitude, it was a longstanding open question to find explicit constructions of point sets and sequences with optimal order of $\mathcal{L}_2$ discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite point sets and recently by the authors of this article for infinite sequences. These constructions can also be extended to give optimal order of the $\mathcal{L}_q$ discrepancy of finite point sets for $q \in (1,\infty)$. The main aim of this article is to give an overview of these constructions and related results

Digital nets in dimension two with the optimal order of LpL_p discrepancy

We study the $L_p$ discrepancy of two-dimensional digital nets for finite $p$. In the year 2001 Larcher and Pillichshammer identified a class of digital nets for which the symmetrized version in the sense of Davenport has $L_2$ discrepancy of the order $\sqrt{\log N}/N$, which is best possible due to the celebrated result of Roth. However, it remained open whether this discrepancy bound also holds for the original digital nets without any modification. In the present paper we identify nets from the above mentioned class for which the symmetrization is not necessary in order to achieve the optimal order of $L_p$ discrepancy for all $p \in [1,\infty)$. Our findings are in the spirit of a paper by Bilyk from 2013, who considered the $L_2$ discrepancy of lattices consisting of the elements $(k/N,\{k \alpha\})$ for $k=0,1,\ldots,N-1$, and who gave Diophantine properties of $\alpha$ which guarantee the optimal order of $L_2$ discrepancy.Comment: 21 page