The motivation for introducing and studying the concept of (t, m, s)-nets and (T, s)- sequences was to generate point sets (also sometimes in high dimensions) with as small a discrepancy as possible. In this chapter we give an overview of theoretical results for the discrepancy of (digital) nets and sequences.

While singular results were already given by Sobol′ [253] and by Faure [68], the first systematic study of the discrepancy of nets was given by Niederreiter [168]. These results can also be found in [177, Chapter 4]. Further results on the star discrepancy of digital nets and sequences, mainly for low dimensions, can be found in [40, 71, 72, 74, 125, 126, 144, 145, 213].

After the work of Niederreiter [172, 177], metrical and average results on the discrepancy of nets and net-sequences were given; see, for instance, [134, 135, 136, 138, 140]. Further, the study of weighted discrepancy of net-type point sets also received considerable attention in recent years (see, for example, [49, 146]).

Even though we have many results for the extreme and star discrepancies, very little is known about concrete theoretical estimates for the Lp-discrepancy, especially for net-type point sets. Singular results in this direction can be found in [20, 22, 75, 142, 143, 212, 244] (results concerning the L2-discrepancy are presented in Chapter 16).

The theory of digital nets and sequences has its roots in uniform distribution modulo one and in numerical integration using quasi–Monte Carlo (QMC) rules. The subject can be traced back to several influential works: the notion of uniform distribution to a classical paper by Weyl [265]; the Koksma–Hlawka inequality, which forms the starting point for analysing QMC methods for numerical integration, to Koksma [121] in the one-dimensional case and to Hlawka [111] in arbitrary dimension. Explicit constructions of digital sequences were first introduced by Sobol′ [253], followed by Faure [68] and Niederreiter [173]. A general principle of these constructions was introduced by Niederreiter in [172], which now forms one of the essential pillars of QMC integration and of this book. These early results are well summarised in references [61, 114, 130, 171 and 177], where much more information on the history and on earlier discoveries can be found.

Since then, numerical integration based on QMC has been developed into a comprehensive theory with many new facets. The introduction of reproducing kernel Hilbert spaces by Hickernell [101] furnished many Koksma–Hlawka-type inequalities. The worst-case integration error can be expressed directly in terms of a reproducing kernel, a function which, together with a uniquely defined inner product, describes a Hilbert space of functions.

Contrary to earlier suppositions, QMC methods are now used for the numerical integration of functions in hundreds or even thousands of dimensions. The success of this approach has been described by Sloan and Woźniakowski in [249], where the concept of weighted spaces was introduced.