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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关于 Korobov p-set 的周期性 $$L_2$$L2 差异的说明

我们研究了$d$维环面中点集的周期性$L_2$-discrepancy。这种差异与移位点集的均方根 $L_2$ 差异、双音概念以及用于在混合平滑的 Sobolev 空间中积分周期函数的体积公式的最坏情况误差密切相关。在差异理论中，许多结果都基于平均参数。为了使这些结果与应用程序相关，需要明确构造具有“平均”差异的点集。在我们的主要结果中，我们研究了 Korobov 的 $p$-sets 并表明该点集具有周期性的 $L_2$-平均阶差异。这个结果与 Novak 和 Woźniakowski 的一个悬而未决的问题有关。