关于Korobov $p$ -集的周期$L_2$ -矛盾的一个注记
A note on the periodic $L_2$-discrepancy of Korobov's $p$-sets
摘要：
本文研究了$d$维环面中点集的周期性$L_2$ -偏差。这种差异与移动点集的均方根$L_2$ -差异密切相关，与diaphony的概念密切相关，与混合光滑的索伯列夫空间中周期函数积分的容积公式的最坏情况误差密切相关。在差异理论中，许多结果是基于平均参数。为了使这样的结果相关的应用程序之一，需要明确的建设点集与“平均”的差异。在我们的主要结果中，我们研究了Korobov的$p$ -集，并证明了这类点集具有平均阶的周期性$L_2$ -偏差。这一结果与Novak和Woonniakowski的一个公开问题有关。
英文摘要：
We study the periodic $L_2$-discrepancy of point sets in the $d$-dimensional torus. This discrepancy is intimately connected with the root-mean-square $L_2$-discrepancy of shifted point sets, with the notion of diaphony, and with the worst case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothness. In discrepancy theory many results are based on averaging arguments. In order to make such results relevant for applications one requires explicit constructions of point sets with ``average'' discrepancy. In our main result we study Korobov's $p$-sets and show that this point sets have periodic $L_2$-discrepancy of average order. This result is related to an open question of Novak and Woźniakowski.
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