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L 2 Discrepancy of Two-Dimensional Digitally Shifted Hammersley Point Sets in Base b

In: Monte Carlo and Quasi-Monte Carlo Methods 2008

Author

Listed:
  • Henri Faure

    (Institut de Mathématiques de Luminy, U.M.R. 6206 CNRS)

  • Friedrich Pillichshammer

Abstract

We give an exact formula for the L 2 discrepancy of two-dimensional digitally shifted Hammersley point sets in base b. This formula shows that for certain bases b and certain shifts the L 2 discrepancy is of best possible order with respect to the general lower bound due to Roth. Hence, for the first time, it is proved that, for a thin, but infinite subsequence of bases b starting with 5,19,71,…, a single permutation only can achieve this best possible order, unlike previous results of White (1975) who needs b permutations and Faure & Pillichshammer (2008) who need 2 permutations.

Suggested Citation

  • Henri Faure & Friedrich Pillichshammer, 2009. "L 2 Discrepancy of Two-Dimensional Digitally Shifted Hammersley Point Sets in Base b," Springer Books, in: Pierre L' Ecuyer & Art B. Owen (ed.), Monte Carlo and Quasi-Monte Carlo Methods 2008, pages 355-368, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-04107-5_22
    DOI: 10.1007/978-3-642-04107-5_22
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