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=12[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(Lα) ≳ (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(Lα) ≳ (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.
|Original language||English (US)|
|Number of pages||27|
|State||Published - Jan 2009|
Bibliographical noteFunding Information:
Acknowledgements. All authors are grateful to the Fields Institute for hospitality and support, and to the National Science Foundation for support. The authors thank the referee for an expert reading, and suggestions that helped to improve the paper.