  We prove an inequality related to questions in Approximation Theory,
Probability Theory, and to Irregularities of Distribution.
  Let $h_R$ denote an $L ^{\infty}$ normalized Haar function adapted to a
dyadic rectangle $R\subset [0,1] ^{3}$. We show that there is a postive $\eta$
so that for all integers $n$, and coefficients $ \alpha (R)$ we have
  2 ^{-n} \sum_{\abs{R}=2 ^{-n}} \abs{\alpha(R)} {}\lesssim{} n ^{1 - \eta}
\NOrm \sum_{\abs{R}=2 ^{-n}} \alpha(R) h_R >.\infty .
  This is an improvement over the `trivial' estimate by an amount of $n ^{-
\eta}$, and the optimal value of $\eta$ (which we do not prove) would be $ \eta
=\frac12$. There is a corresponding lower bound on the $L ^{\infty}$ norm of
the Discrepancy function of an arbitary distribution of a finite number of
points in the unit cube in three dimensions. The prior result, in dimension 3,
is that of J{\'o}zsef Beck \cite{MR1032337}, in which the improvement over the
trivial estimate was logarithmic in $n$. We find several simplifications and
extensions of Beck's argument to prove the result above.
