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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

##### MSC:
 11K06 General theory of distribution modulo $$1$$ 65C10 Random number generation in numerical analysis
##### Keywords:
pseudo-random numbers; discrepancy