Irregularities of distributions and extremal sets in combinatorial complexity theory

TL;DR

This paper provides an elementary combinatorial proof for a known lower bound on the star-discrepancy of point sets in high-dimensional cubes, linking irregularities of distributions to discrepancy measures.

Contribution

It offers a new combinatorial proof of a discrepancy bound and characterizes irregular point sets with high star-discrepancy.

Findings

Elementary combinatorial proof of discrepancy bound

Identification of irregular point sets with large discrepancy

Characterization of distribution irregularities in high dimensions

Abstract

In 2004 the second author of the present paper proved that a point set in $[0,1]^d$ which has star-discrepancy at most $\varepsilon$ must necessarily consist of at least $c_{abs} d \varepsilon^{-1}$ points. Equivalently, every set of $n$ points in $[0,1]^d$ must have star-discrepancy at least $c_{abs} d n^{-1}$. The original proof of this result uses methods from Vapnik--Chervonenkis theory and from metric entropy theory. In the present paper we give an elementary combinatorial proof for the same result, which is based on identifying a sub-box of $[0,1]^d$ which has approximately $d$ elements of the point set on its boundary. Furthermore, we show that a point set for which no such box exists is rather irregular, and must necessarily have a large star-discrepancy.