On some remarkable properties of the two-dimensional Hammersley point set in base 2

Kritzer, Peter

doi:10.5802/jtnb.540

Kritzer, Peter ¹

¹ Department of Mathematics University of Salzburg Hellbrunnerstr. 34 A-5020 Salzburg, Austria

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 203-221.

Résumé
Abstract

Nous examinons une classe spéciale de $(0, m, 2)$ -réseaux en base 2. Particulièrement, nous nous occupons du réseau de Hammersley en deux dimensions qui joue un rôle spécial parmi ce type de réseaux, puisque nous démontrons que c’est le plus mal distribué quant à la discrépance à l’origine. En le montrant, nous améliorons un majorant connu pour la discrépance à l’origine de $(0, m, 2)$ -réseaux en base 2. De plus, nous démontrons qu’on peut obtenir des réseaux avec une discrépance à l’origine très basse en transformant le réseau de Hammersley d’une manière appropriée.

We study a special class of $(0, m, 2)$ -nets in base 2. In particular, we are concerned with the two-dimensional Hammersley net that plays a special role among these since we prove that it is the worst distributed with respect to the star discrepancy. By showing this, we also improve an existing upper bound for the star discrepancy of digital $(0, m, 2)$ -nets over $ℤ_{2}$ . Moreover, we show that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.

MR Zbl

DOI : 10.5802/jtnb.540

Affiliations des auteurs :

Kritzer, Peter ¹

¹ Department of Mathematics University of Salzburg Hellbrunnerstr. 34 A-5020 Salzburg, Austria

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Kritzer, Peter. On some remarkable properties of the two-dimensional Hammersley point set in base 2. Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 203-221. doi : 10.5802/jtnb.540. https://www.numdam.org/articles/10.5802/jtnb.540/

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