A note on the periodic $L_2$-discrepancy of Korobov's $p$-sets

Abstract

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 Wo\'{z}niakowski.