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

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

2006-09-28

arxiv.org/abs/math/0609817v2

Mathematics

Number Theory

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

Scientific paper

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}.

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