Discrepancy of point sequences on fractal sets

We consider asymptotic bounds for the discrepancy of point sets on a class of fractal sets. By a method of R. Alexander, we prove that for a wide class of fractals, the L-2-discrepancy (and consequently also the worst-case discrepancy) of an N-point set with respect to halfspaces is at least of the order N-1/2-1/2s, where s is the Hausdorff dimension of the fractal. We also show that for many fractals, this bound is tight for the L-2-discrepancy. Determining the correct order of magnitude of the worst-case discrepancy remains a challenging open problem.