Pseudozufallszahlen und die Theorie der Gleichverteilung. (Pseudo-random numbers and the theory of uniform distribution). (German) Zbl 0616.10040

Let \({\mathfrak x}_ 0,...,{\mathfrak x}_{N-1}\) be a set of N points in \([0,1]^ s\), \(s\geq 1\). The discrepancy of the set (\({\mathfrak x}_ i)\) is defined by \(D_ N=\sup_{J}| A(J;N)/N-Vol(J)|,\) where J denotes all intervals \(J=[0,t_ 1]\times... \times [0,t_ s]\subseteq [0,1]^ s\) and A(J;N) is the number of n such that \(0\leq n<N\) and \(x_ n\in J\). It is known that if f(\({\mathfrak t})\) satisfies certain conditions, then \(\int_{[0,1]^ s}f({\mathfrak t}) d{\mathfrak t}-(1/N)\sum^{N- 1}_{n=0}f({\mathfrak x}_ n)\) is bounded by \(D_ N\). In this paper the author gives estimations for upper and lower bounds of \(D_ N\) for various sets of points in \([0,1]^ s\).
Reviewer: Wang Yuan


11K06 General theory of distribution modulo \(1\)
65C10 Random number generation in numerical analysis