  Let A_N be an N-point distribution in the unit square in the Euclidean plane.
We consider the Discrepancy function D_N(x) in two dimensions with respect to
rectangles with lower left corner anchored at the origin and upper right corner
at the point x. This is the difference between the actual number of points of
A_N in such a rectangle and the expected number of points - N x_1x_2 - in the
rectangle. We prove sharp estimates for the BMO norm and the exponential
squared Orlicz norm of D_N(x). For example we show that necessarily
||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On the other
hand we use a digit scrambled version of the van der Corput set to show that
this bound is tight in the case N=2^n, for some positive integer n. These
results unify the corresponding classical results of Roth and Schmidt in a
sharp fashion.
