Exponential Squared Integrability for the Discrepancy Function in Two Dimensions

TL;DR

This paper establishes sharp bounds on the discrepancy function in two dimensions, showing that the exponential squared integrability norm grows at least as the square root of log N, and demonstrates tightness using a digit scrambled van der Corput set.

Contribution

It provides sharp estimates for the BMO and exponential squared Orlicz norms of the discrepancy function, unifying classical results of Roth and Schmidt.

Findings

Lower bound: ||D_N||_(expL^2) > c(logN)^(1/2)

Upper bound tightness shown for N=2^n using digit scrambled van der Corput set

Results unify and sharpen classical discrepancy bounds

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.