Component-by-Component Construction of Hybrid Point Sets Based on Hammersley and Lattice Point Sets

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.KeywordsStar DiscrepancyActa ArithHybrid SequenceMonatsh MathHammersley PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.