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The weighted star discrepancy of Korobov's $ p$-sets


Authors: Josef Dick and Friedrich Pillichshammer
Journal: Proc. Amer. Math. Soc. 143 (2015), 5043-5057
MSC (2010): Primary 11K38, 65C05
Published electronically: May 7, 2015
MathSciNet review: 3411125
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Abstract: We analyze the weighted star discrepancy of so-called $ p$-sets which go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have largely been ignored since a number of other constructions have been discovered which achieve a better convergence rate. However, it has recently been discovered that the $ p$-sets perform well in terms of the dependence on the dimension.

We prove bounds on the weighted star discrepancy of the $ p$-sets which hold for any choice of weights. For product weights, we give conditions under which the discrepancy bounds are independent of the dimension $ s$. This implies strong polynomial tractability for the weighted star discrepancy. We also show that a very weak condition on the product weights suffices to achieve polynomial tractability.


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Additional Information

Josef Dick
Affiliation: School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia
Email: josef.dick@unsw.edu.au

Friedrich Pillichshammer
Affiliation: Institut für Analysis, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria
Email: friedrich.pillichshammer@jku.at

DOI: https://doi.org/10.1090/proc/12636
Received by editor(s): March 31, 2014
Received by editor(s) in revised form: September 11, 2014
Published electronically: May 7, 2015
Additional Notes: The first author was supported by a QEII Fellowship of the Australian Research Council DP1097023.
The second author was supported by the Austrian Science Fund (FWF): Project F5509-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society