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On the L p discrepancy of two-dimensional folded Hammersley point sets

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Abstract

We give an explicit construction of two-dimensional point sets whose L p discrepancy is of best possible order for all \({1 \le p \le \infty}\). It is provided by folding Hammersley point sets in base b by means of the b-adic baker’s transformation which has been introduced by Hickernell (Monte Carlo and quasi-Monte Carlo methods. Springer, Berlin, 274–289, 2002) for b =  2 and Goda, Suzuki, and Yoshiki (The b-adic baker’s transformation for quasi-Monte Carlo integration using digital nets. arXiv:1312.5850 [math:NA], 2013) for arbitrary \({b \in \mathbb{N}}\), \({b \ge 2}\). We prove that both the minimum Niederreiter–Rosenbloom–Tsfasman weight and the minimum Dick weight of folded Hammersley point sets are large enough to achieve the best possible order of L p discrepancy for all \({1 \le p \le \infty}\).

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  1. Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Takashi Goda

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  1. Takashi Goda
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Goda, T. On the L p discrepancy of two-dimensional folded Hammersley point sets. Arch. Math. 103, 389–398 (2014). https://doi.org/10.1007/s00013-014-0698-1

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  • DOI: https://doi.org/10.1007/s00013-014-0698-1

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