On the small ball inequality in three dimensions

Abstract

Let hR 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 2-n∑|R|=2-n|α(R)|≲n1-η‖∑|R|=2-nα(R)hR‖∞. 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