论文标题
极端和周期性的$ L_2 $平面点的差异
Extreme and periodic $L_2$ discrepancy of plane point sets
论文作者
论文摘要
在本文中，我们研究了飞机点集的极端和周期性$ L_2 $差异。极端差异基于任意矩形作为测试集，而周期性差异使用“周期间隔”，可以看作是圆环的间隔。周期性$ L_2 $差异最多是乘法因素，也称为digion。主要结果是Hammersley Point Set和Rational Lattices的这类差异的确切公式。 为了重视所获得的结果，我们还证明了在尺寸$ d $中的任意点集的极端$ L_2 $差异上的一般下限，这是$（\ log n）^{（d-1）/2} $，例如标准和周期性$ L_2 $差异。我们的结果证实，Hammersley Point Set的极端和周期性$ L_2 $差异是最好的渐近数量级。这与Hammersley Point Set的标准$ L_2 $差异相反。此外，我们的确切公式还表明，斐波那契晶格的$ L_2 $差异是最佳订单。 我们还证明，极端的$ L_2 $差异始终由标准的$ L_2 $差异主导，这一结果已经由Morokoff和Caflisch引入了1994年的Extreme $ L_2 $差异的概念。
In this paper we study the extreme and the periodic $L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the torus. The periodic $L_2$ discrepancy is, up to a multiplicative factor, also known as diaphony. The main results are exact formulas for these kinds of discrepancies for the Hammersley point set and for rational lattices. In order to value the obtained results we also prove a general lower bound on the extreme $L_2$ discrepancy for arbitrary point sets in dimension $d$, which is of order of magnitude $(\log N)^{(d-1)/2}$, like the standard and periodic $L_2$ discrepancies, respectively. Our results confirm that the extreme and periodic $L_2$ discrepancies of the Hammersley point set are of best possible asymptotic order of magnitude. This is in contrast to the standard $L_2$ discrepancy of the Hammersley point set. Furthermore our exact formulas show that also the $L_2$ discrepancies of the Fibonacci lattice are of the optimal order. We also prove that the extreme $L_2$ discrepancy is always dominated by the standard $L_2$ discrepancy, a result that was already conjectured by Morokoff and Caflisch when they introduced the notion of extreme $L_2$ discrepancy in the year 1994.
