On the Discrepancy Function in Arbitary Dimension, Close to L ^{1}
Abstract
Description
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
17 pages. To appear in Analysis Mathematica. Many changes, and an additional section on Hardy space and the Discrepancy function