  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.
