The Supremum Norm of the Discrepancy Function: Recent Results and Connections

TL;DR

This paper surveys recent advances in understanding the lower bounds of the discrepancy function's supremum norm in higher dimensions, highlighting new results, methods, and connections to other mathematical fields.

Contribution

It reviews recent improvements on lower bounds of the discrepancy function in dimensions three and higher, and explores their links to probability, approximation theory, and analysis.

Findings

Average case bound of (log N)^{(d-1)/2} for the L-infty norm.

Partial improvements to Roth's exponent in higher dimensions.

Connections between discrepancy bounds and other mathematical disciplines.

Abstract

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in higher dimensions have been established by the authors and Armen Vagharshakyan. We survey these results, the underlying methods, and some of their connections to other subjects in probability, approximation theory, and analysis.