Let AN be an N-point set in the unit square and consider the discrepancy function DN(x):= #(AN ∩ [0, x)) - N|(0, x)|, where x = (x1, X2) ∈ [0, 1]2, [0, x) = Πt=1 2[0, xt), and |[0, x)| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of Schmidt [Irregularities of distribution. VII. Acta Arith. 21 (1972), 45-50] on the L∞ norm of DN. We show that necessarily ||DN||exp(L α) ≳ (log N)1-1/α, 2 ≤ α < ∞. The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we have ||DN|| exp(L α) ≳ (log N)1-1/α, 2 ≤ α < ∞., whenever N = 2n for some positive integer n. This estimate depends upon variants of the Chang-Wilson - Wolff inequality [S.-Y. A. Chang, J. M. Wilson and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60(2) (1985), 217-246]. We also provide similar estimates for the BMO norm of DN.
2009. Vol. 55, no 1-2, 1-27 p.