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

Henri Faure () and Friedrich Pillichshammer ()
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Henri Faure: Institut de Mathématiques de Luminy, U.M.R. 6206 CNRS

A chapter in Monte Carlo and Quasi-Monte Carlo Methods 2008, 2009, pp 355-368 from Springer

Abstract: 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.

Keywords: Exact Formula; Implied Constant; Arbitrary Basis; Single Shift; Probabilistic Number Theory (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-04107-5_22

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DOI: 10.1007/978-3-642-04107-5_22

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