Exponential Squared Integrability for the Discrepancy Function in Two Dimensions

Abstract

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.

27 pages, 3 figures. Many improvements reflecting the comments and observations of the referee. Final version. Submitted to Mathematika

Cited by in corpus (5)

• Calculation of Discrepancy Measures and Applications
• The Supremum Norm of the Discrepancy Function: Recent Results and Connections
• Optimal $L_p$-discrepancy bounds for second order digital sequences
• Tractability properties of the discrepancy in Orlicz norms
• The BMO-discrepancy suffers from the curse of dimensionality