A lower bound for the dispersion on the torus

TL;DR

This paper establishes a fundamental lower bound on the volume of empty axis-parallel boxes in the torus, which also applies to discrepancy, providing insights into uniform point distributions in high-dimensional spaces.

Contribution

It proves a universal lower bound for the largest empty box volume and discrepancy on the torus, applicable to all point sets regardless of their configuration.

Findings

Lower bound of min{1, d/n} for empty box volume

Lower bound applies to discrepancy on the torus

Results hold for all dimensions and point sets

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.