Irregularities of distributions and extremal sets in combinatorial complexity theory

In 2004 the second author of the present paper proved that a point set in $[0,1{]}^d$ which has star-discrepancy at most $\epsilon$ must necessarily consist of at least $c_{abs}d{\epsilon }^{-1}$ 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.