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 positive $\eta \lt 1/2$ so that for all integers $n$ and coefficients $\alpha (R)$ , we have \[2^{-n} \sum_{|R|=2^{-n}} |{\alpha(R)}| \lesssim n ^{1 - \eta} \|\sum_{|R|=2^{-n}} \alpha(R) h_R \|_\infty . \] This is an improvement over the trivial estimate by an amount of $n ^{-\eta}$ , while the small ball conjecture says that the inequality should hold with $\eta=1/2$ . There is a corresponding lower bound on the $L^{\infty}$ -norm of the discrepancy function of an arbitrary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension three, is that of József Beck [1, Theorem 1.2], 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