A lower bound for the dispersion on the torus
arXiv:1510.0461726 citations
Originality Incremental advance
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Provides a fundamental theoretical lower bound for dispersion on the torus, relevant to mathematicians and computer scientists studying point sets and discrepancy.
The paper proves a lower bound of min{1, d/n} for the volume of the largest empty axis-parallel box in the d-dimensional torus for any set of n points, which also implies a lower bound for discrepancy.
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.