On the small ball inequality in three dimensions

Let h_R denote an L^∞-normalized Haar function adapted to a dyadic rectangle R ⊂ [0, 1]³. We show that there is a positive η < 1/2 so that for all integers n and coefficients α(R), we have (Equation Presented) This is an improvement over the trivial estimate by an amount of n^-η, while the small ball conjecture says that the inequality should hold with η = 1/2. There is a corresponding lower bound on the L^∞-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.