平面点集的极值和周期$L_2$差异
Extreme and periodic $L_2$ discrepancy of plane point sets
摘要：
本文研究了平面点集的极值和周期性$L_2$差。极端差异是基于任意矩形作为测试集，而周期性差异使用“周期性间隔”，这可以被视为环面上的间隔。周期性的$L_2$差异，直到一个乘法因子，也被称为diaphony。主要结果是精确的公式，这些种差异的Hammersley点集和合理的格。为了评估所得到的结果，我们还证明了一个一般的下界上的极端$L_2$差异为任意点集的尺寸$d$，这是数量级$(\log N)^{(d-1)/2}$，像标准和周期性的$L_2$差异，分别。我们的结果证实了极端和周期性的$L_2$差异的Hammersley点集是最好的可能渐近数量级。这与Hammersley点集的标准$L_2$差异形成对比。此外，我们的精确公式表明，也$L_2$差异的Fibonacci格是最佳的顺序。我们还证明了极端的$L_2$差异总是由标准的$L_2$差异所主导，这一结果已经由Morokoff和Caflisch在1994年引入极端$L_2$差异的概念时所证实。
英文摘要：
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.
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