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On an example of finite hybrid quasi-Monte Carlo point sets. (English) Zbl 1271.11080

The author considers finite hybrid point sets in the unit cube. In this particular case he considers \(s+d+1\)-dimensional point sets \(S_N=(\mathbf x_n,\mathbf y_n)_{n=0}^{N}\), where the \(\mathbf x_n\) form \(s+1\)-dimensional Hammersley point sets and the \(\mathbf y_n\) form \(d\)-dimensional lattice point sets of the form \(\mathbf{y}_n=\left\{\frac{n\mathbf g}{N}\right\}\), where \(\mathbf g\in\{1,\dots,N\}^d\). As a main result the author shows that there exists a vector \(\mathbf g\) such that \[ D_N^*(S_N)=\mathcal O\left( \frac{(\log N)^{s+d+1}}{N}\right), \] with the implied constant independent of \(N\).
Furthermore the author shows that \(\mathbf g\) can be chosen as \((g,g^2,\dots,g^d)\mod N\) for some \(g\), provided that \(N\) is a prime different from the primes \(p_1,\dots,p_s\) that are a basis of the Hammersley point set \((\mathbf x_n)_{n=0}^N\).

MSC:

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