On the small ball inequality in three dimensions

Abstract

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 $\eta <1/2$ so that for all integers $n$ and coefficients $\alpha (R)$, we have ${\displaystyle 2^{-n}\sum _{\left|R\right|=2^{-n}}|\alpha (R)|\lesssim n^{1-\eta }\Vert \sum _{\left|R\right|=2^{-n}}\alpha (R)h_R{\Vert }_{\infty }.}$ 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