A lower bound for the dispersion on the torus

Abstract

We consider the volume of the largest axis-parallel box in the $d$-dimensional torus that contains no point of a given point set ${\mathcal{P}}_n$ with $n$ elements. We prove that, for all natural numbers $d,n$ and every point set ${\mathcal{P}}_n$, this volume is bounded from below by $\min \{1,d/n\}$. This implies the same lower bound for the discrepancy on the torus.

6 pages

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• On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points