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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The weighted star discrepancy of Korobov’s $p$-sets
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by Josef Dick and Friedrich Pillichshammer PDF
Proc. Amer. Math. Soc. 143 (2015), 5043-5057 Request permission

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
  • MR Author ID: 661956
  • ORCID: 0000-0001-6952-9218
  • Email: friedrich.pillichshammer@jku.at
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 5043-5057
  • MSC (2010): Primary 11K38, 65C05
  • DOI: https://doi.org/10.1090/proc/12636
  • MathSciNet review: 3411125