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A lower bound on a quantity related to the quality of polynomial lattices. (English) Zbl 1247.11097
For $$b$$ be a prime, let $$\mathbb{F}_b$$ be a finite field consisting of $$b$$ elements. Let $$f\in \mathbb{F}_b[x]$$ be a polynomial over $$\mathbb{F}_b$$. For a vector $$\mathbf{g}=(g_1,\ldots, g_s)\in\mathbb{F}_b[x]^s$$, let $$P(\mathbf{g},f)$$ be a polynomial lattice with $$N=b^m$$ points in dimension $$s$$. Let $$G_{b,m}=\{a\in\mathbb{F}_b[x]: \deg(a)<m\}$$. A quantity $$R_b(\mathbf{g},f)$$ is used for studying the quality of $$P(\mathbf{g},f)$$. It was shown by H. Niederreiter [Random number generation and quasi-Monte Carlo methods. Philadelphia, PA: SIAM (1992; Zbl 0761.65002)] that the star discrepancy of $$P(\mathbf{g},f)$$ satisfies $$D_N^*(P(\mathbf{g},f))\leq s/N+R_b(\mathbf{g},f)$$ and there exists a vector $$\mathbf{g}\in G_{b,m}^s$$ such that $$R_b(\mathbf{g},f)\leq C_{s,b}m^s/b^m$$ for some $$C_{s,b}>0$$. These results yield the existence of $$P(\mathbf{g},f)$$ with $$D_N^*(P(\mathbf{g},f))=O((\log N)^s/N)$$. In the paper under review, the authors prove that Niederreiter’s upper bound for $$R_b(\mathbf{g},f)$$ is essentially best possible.

##### MSC:
 11K06 General theory of distribution modulo $$1$$ 11K38 Irregularities of distribution, discrepancy
##### Keywords:
polynomial lattices; star discrepancy; digital nets
Zbl 0761.65002
Full Text:
##### References:
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