On the Small Ball Inequality in All Dimensions

TL;DR

This paper establishes a new lower bound on the supremum norm of hyperbolic sums of Haar functions in all dimensions, improving average case bounds and advancing understanding in irregularity, approximation, and probability theories.

Contribution

It provides the first non-trivial lower bounds for the small ball inequality in dimensions four and higher, extending previous results from dimension three.

Findings

Improved lower bounds on L^{ty} norms of Haar function sums in high dimensions

Extension of small ball inequality results to dimensions 4 and above

New implications for irregularity of distributions, approximation, and probability conjectures

Abstract

Let h_R denote an L ^{\infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{\infty} norm of the `hyperbolic' sums $$ \sum _{|R|=2 ^{-n}} \alpha(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^{\eta} for some positive \eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.