\(L_2\) discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations. (English) Zbl 1198.11072

Faure, Henri; Pillichshammer, Friedrich; Pirsic, Gottlieb; Schmid, Wolfgang Ch.

Acta Arith. 141, No. 4, 395-418 (2010).

Let \(b\geq 2\) be an integer, let \(\mathfrak{S}_b\) be the set of all permutations of \(\{0, 1, \ldots, b-1\}\), and let \(\Sigma=(\sigma_0,\ldots, \sigma_{n-1})\in \mathfrak{S}_b^n\). For an integer \(1\leq N\leq b^n\), write \(N-1=\sum_{r=0}^{n-1}a_r(N)b^r\) in the \(b-\)adic system and define \[ S_b^{\Sigma}(N):=\sum_{n=0}^{n-1}\frac{\sigma_r(a_r(N))}{b^{r+1}}. \] The generalized two-dimensional Hammersley point set in base \(b\) with \(\Sigma\) is defined by \[ \mathcal{H}_{b,n}^{\Sigma}:=\left\{\left(S_{b}^{\Sigma}(N),\frac{N-1}{b^n}\right):1\leq N\leq b^n\right\}. \] Let \(\tau\in \mathfrak{S}_b\) be given by \(\tau(k)=b-1-k\) and let \(\bar{\sigma}:=\tau\circ\sigma\) for \(\sigma\in \mathfrak{S}_b\). For \(\Sigma\in\{\sigma, \bar{\sigma}\}^n\) the authors show that the \(L_2-\)discrepancy of \(\mathcal{H}_{b,n}^{\Sigma}\) has the optimal upper bound. Furthermore, for \(\sigma\in\mathfrak{S}_b\) with \(\sigma\circ\tau=\tau\circ\sigma\) and \(\Sigma\in\{\sigma, \bar{\sigma}\}^n\) they give an exact formula for the \(L_2-\)discrepancy of \(\mathcal{H}_{b,n}^{\Sigma}\).

Reviewer: Yukio Ohkubo (Kagoshima)

MSC:

11K06	General theory of distribution modulo \(1\)
11K38	Irregularities of distribution, discrepancy

Keywords:

\(L_2\) discrepancy; generalized Hammersley point set