  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}.
