Irregularities of Distribution

Abstract

This book is an authoritative description of the various approaches to and methods in the theory of irregularities of distribution. The subject is primarily concerned with number theory, but also borders on combinatorics and probability theory. The work is in three parts. The first is concerned with the classical problem, complemented where appropriate with more recent results. In the second part, the authors study generalizations of the classical problem, pioneered by Schmidt. Here, they include chapters on the integral equation method of Schmidt and the more recent Fourier transform technique. The final part is devoted to Roth's '1/4-theorem'.

Test Coverage for Continuous and Hybrid Systems

“…We first briefly recall the star discrepancy. The star discrepancy is an important notion in equidistribution theory as well as in quasi-Monte Carlo techniques (see for example [1]). Recently, it was also used in probabilistic motion planning to enhance the sampling uniformity [3].…”

“…If a staying set I q is not a box, we can take the smallest oriented box that encloses it, and apply the star discrepancy definition in (1) to that box after an appropriate coordination change. We can see that a large value of Cov(P) indicates a good space-covering quality.…”

On the discrepancy function in arbitrary dimension, close to L 1

“…We recall some definitions and facts about Discrepancy which are well represented in the literature, (Roth, 1954;Schmidt, 1977a;Beck and Chen, 1987).…”

On the mean square weighted L2discrepancy of randomized digital (t,m,s)-nets over Z2

“…. , x 2 m −1 } be a digital (t, m, 1)-net over Z 2 generated by the matrix C. Let 1 2 + x n,2 2 2 + · · · be the dyadic digit expansion of x n . In [5] a randomization method was considered which uses a digital shift σ = σ 1 /2 + σ 2 /2 2 + · · ·, where σ ∈ [0, 1) was chosen randomly.…”

“…. , z 2 m −1 } is given by z n = z n, 1 2 + · · · + z n,m 2 m + δ n . This means we apply the same digital shift to the first m digits, whereas the following digits are shifted independently for each x n .…”