  The discrepancy function measures the deviation of the empirical distribution
of a point set in $[0,1]^d$ from the uniform distribution. In this paper, we
study the classical discrepancy function with respect to the BMO and
exponential Orlicz norms, as well as Sobolev, Besov and Triebel-Lizorkin norms
with dominating mixed smoothness. We give sharp bounds for the discrepancy
function under such norms with respect to infinite sequences.
