Nets, (t, s)-Sequences, and Algebraic Geometry

Abstract

The star discrepancy is a classical measure for the irregularity of distribution of finite sets and infinite sequences of points in the s-dimensional unit cube P = [0, 1]8. Point sets and sequences with small star discrepancy in I8 are informally called low-discrepancy point sets, respectively low-discrepancy sequences, in I. It is also customary to speak of sets, respectively sequences, of quasirandom points in 78. Such point sets and sequences play a crucial role in applications of numerical quasi-Monte Carlo methods. In fact, the efficiency of a quasi-Monte Carlo method depends to a significant extent on the quality of the quasirandom points that are employed, i.e., on how small their star discrepancy is. Therefore, it is a matter of considerable interest to devise techniques for the construction of point sets and sequences with as small a star discrepancy as possible. The reader who desires more background on discrepancy theory and quasi-Monte Carlo methods is referred to the books of Hua and Wang [9], Kuipers and Niederreiter [10], and Niederreiter[21], the survey article of Niederreiter [16], and the recent monograph of Drmota and Tichy [4].