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Dyadic Diaphony of Digital Nets Over ℤ2

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Abstract.

The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ2. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the \({\cal L}_2\) discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.

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Authors and Affiliations

  1. University of New South Wales, Sydney, Australia

    Josef Dick

  2. Universität Linz, Austria

    Friedrich Pillichshammer

Authors
  1. Josef Dick
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  2. Friedrich Pillichshammer
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Additional information

The first author is supported by the Australian Research Council under its Center of Excellence Program.

The second author is supported by the Austrian Research Foundation (FWF), Project S 8305 and Project P17022-N12.

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Dick, J., Pillichshammer, F. Dyadic Diaphony of Digital Nets Over ℤ2. Mh Math 145, 285–299 (2005). https://doi.org/10.1007/s00605-004-0287-7

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  • DOI: https://doi.org/10.1007/s00605-004-0287-7

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