On the Discrepancy Function in Arbitary Dimension, Close to L ^{1}

Abstract

Let ${\mathcal{A}}_N$ to be $N$ points in the unit cube in dimension $d$, and consider the Discrepency function D_N(\vec x) \coloneqq \sharp \mathcal A_N \cap [\vec 0,\vec x)-N \abs{[\vec 0,\vec x)} Here, $\vec{x}=(x_1,...c,x_d)$ and $[0,\vec{x})={\prod }_{t=1}^d[0,x_t)$. We show that necessarily \norm D_N. L ^{1} (\log L) ^{(d-2)/2}. \gtrsim (\log N) ^{d/2} . In dimension $d=2$, the `$\log L$' term has power zero, which corresponds to a Theorem due to \cite{MR637361}.

17 pages. To appear in Analysis Mathematica. Many changes, and an additional section on Hardy space and the Discrepancy function

References in corpus (1)

• On the Small Ball Inequality in All Dimensions