Abstract
In a series of recent articles, such as, e.g., (Hellekalek (2012) Hybrid function systems in the theory of uniform distribution of sequences. Monte Carlo and quasi-Monte Carlo methods 2010. Springer, Berlin, pp. 435–450; Hofer, Kritzer, Larcher, Pillichshammer (Int J Number Theory 5:719–746, 2009); Kritzer (Monatsh Math 168:443–459, 2012); Niederreiter (Acta Arith 138:373–398, 2009)), point sets mixed from integration node sets in different sorts of quasi-Monte Carlo rules have been studied. In particular, a finite version, based on Hammersley and lattice point sets, was introduced in Kritzer (Monatsh Math 168:443–459, 2012), where the existence of such hybrid point sets with low star discrepancy was shown. However, up to now it has remained an open problem whether such low discrepancy hybrid point sets can be explicitly constructed. In this paper, we solve this problem and discuss component-by-component constructions of the desired point sets.
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Acknowledgements
P. Kritzer gratefully acknowledges the support of the Austrian Science Fund (FWF), Project P23389-N18. G. Leobacher gratefully acknowledges the support of the Austrian Science Fund (FWF), Project P21196. F. Pillichshammer is partially supported by the Austrian Science Fund (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.
The authors would like to thank Josef Dick, Frances Y. Kuo and Ian H. Sloan for their hospitality during the authors’ stay at the School of Mathematics and Statistics at the University of New South Wales in February 2012, where parts of this paper were written. Furthermore, the authors gratefully acknowledge the support of the Australian Research Council.
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Kritzer, P., Leobacher, G., Pillichshammer, F. (2013). Component-by-Component Construction of Hybrid Point Sets Based on Hammersley and Lattice Point Sets. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_25
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