Exponential Squared Integrability for the Discrepancy Function in Two Dimensions

الأصل · EN

Let Aₙ be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function Dₙ(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 Aₙ in such a rectangle and the expected number of points - N x₁x₂ - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of Dₙ(x). For example we show that necessarily ||Dₙ||(expL²) >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=2ⁿ, for some positive integer n. These results unify the corresponding classical results of Roth and Schmidt in a sharp fashion.