On some remarkable properties of the two-dimensional Hammersley point set in base 2. (English) Zbl 1103.11024

Kritzer, Peter

J. Théor. Nombres Bordx. 18, No. 1, 203-221 (2006).

Let \(H=\{\mathbf{x}_0, \ldots , \mathbf{x}_{2^m-1}\}\) be the \((0,m,2)\)-Hammersley net over \(\mathbb{Z}_2\), where \(\mathbb{Z}_2\) is the finite field with two elements. Let \(Y=\{\mathbf{y}_0, \ldots , \mathbf{y}_{2^m-1}\}\) be the net that is obtained by shifting a fixed digital \((0,m,2)\)-net over \(\mathbb{Z}_2\) in the second coordinate by an arbitrary vector \(\vec{\sigma}\) in \(\mathbb{Z}_2^m\). The author proves that
\[ D_N^*(Y)\leq D_N^*(H)=\left(\frac m3 + \frac {13}9 - (-1)^m\cdot \frac{4}{9\cdot 2^m}\right)2^{-m}, \]
where \(N=2^m\) and \(D_N^*\) is the star discrepancy. This result improves an upper bound for the star discrepancy of digital \((0,m,2)\)-nets over \(\mathbb{Z}_2\) by G. Larcher and F. Pillichshammer [Acta Arith. 106, No. 4, 379–408 (2003; Zbl 1054.11039)]. This result also implies that any digital \(m\)-bit shift applied to \(H\) cannot have negative effects on the star discrepancy. In fact, it is shown that any digital shift different from \(\vec{0}\) of the Hammersley net results in a real improvement of the star discrepancy. Furthermore, the author shows that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.

Reviewer: Yukio Ohkubo (Kagoshima)

MSC:

11K38	Irregularities of distribution, discrepancy
11K31	Special sequences

Keywords:

Hammersley point set; star discrepancy; digital \((0; m; 2)\)-nets; digital shifts

Citations:

Zbl 1054.11039

References:

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