On the Small Ball Inequality in Three Dimensions

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.