  We analyze the weighted star discrepancy of so-called $p$-sets which go back
to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since
then, these sets have largely been ignored since a number of other
constructions have been discovered which achieve a better convergence rate.
However, it has recently been discovered that the $p$-sets perform well in
terms of the dependence on the dimension.
  We prove bounds on the weighted star discrepancy of the $p$-sets which hold
for any choice of weights. For product weights we give conditions under which
the discrepancy bounds are independent of the dimension $s$. This implies
strong polynomial tractability for the weighted star discrepancy. We also show
that a very weak condition on the product weights suffices to achieve
polynomial tractability.
