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Low-discrepancy and low-dispersion sequences. (English) Zbl 0651.10034

Let \({\mathfrak x}_ 1,{\mathfrak x}_ 2,..\). be a sequence of points in the unit cube \(I^ s=[0,1)^ s\), \(s\geq 1\). Let J be a subinterval of I and \(D(J;N)=A(J;N)-V(J)N\), where A(J;N) is the number of n, \(1\leq n\leq N\), with \({\mathfrak x}_ n\in J\) and V(J) is the volume of J. Then \(\Delta (N)=\sup_{J} | D(J;N)|\) is called the discrepancy of the first N terms of the sequence \({\mathfrak x}_ 1,{\mathfrak x}_ 2...\). Halton first constructed a low-discrepancy sequence such that \(\Delta (N)\leq c_ s(\log N)^ s+O(c \log N)^{s-1},\quad N\geq 2.\) The author obtains sequences in \(I^ s\) based on the theory of (t,s)-sequences with smallest constant \(c_ s\) that is currently known.
Reviewer: Wang Yuan

MSC:

11K38 Irregularities of distribution, discrepancy
11K06 General theory of distribution modulo \(1\)
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