# On the Small Ball Inequality in Three Dimensions

#### Abstract

|R|=2−n |α(R)| . n1−η ∥∥∥ ∑ |R|=2−n α(R)hR ∥∥∥∞ This is an improvement over the ‘trivial’ estimate by an amount of n−η, and the optimal value of η (which we do not prove) would be η = 12 . There is a corresponding lower bound on the L∞ 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 József Beck [1], 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.

### Cite this paper

@inproceedings{Bilyk2006OnTS, title={On the Small Ball Inequality in Three Dimensions}, author={Dmitriy Bilyk and ANDMICHAEL T. LACEY and Michael T. Lacey}, year={2006} }