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The dispersion of the Hammersley Sequence in the unit square

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Abstract

The notion of dispersion, a measure of denseness of sequences, plays an important role in quasi-Monte Carlo optimization. In this paper, we obtain an explicit formula for the dispersion of an important low dispersion sequence, namely the Hammersley Sequence in the unit square. The dispersiond M of theM points of this sequence, whereM=2N withN a positive integer is given by

$$d_M = \frac{{\sqrt {2M - 2\sqrt M + 1} }}{M},if N is even, d_M = \frac{{\sqrt {\left( {5/2} \right)M - \sqrt {8M} + 1} }}{M},if N is odd.$$

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Authors and Affiliations

  1. Department of Mathematics, University of the West Indies, Kingston 7, Jamaica, West Indies

    Paul Peart

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  1. Paul Peart
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Additional information

This paper is derived from the author's Ph. D. program under the supervision of ProfessorH. Niederreiter at the University of the West Indies. The author wishes to express his gratitude to ProfessorNiederreiter for his kind helpfulness and encouragement.

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Peart, P. The dispersion of the Hammersley Sequence in the unit square. Monatshefte für Mathematik 94, 249–261 (1982). https://doi.org/10.1007/BF01295787

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  • DOI: https://doi.org/10.1007/BF01295787

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