A note on the dispersion of admissible lattices

TL;DR

This paper demonstrates that the volume of axis-parallel boxes avoiding an admissible lattice in -dimensional space is uniformly bounded, leading to optimal dispersion order for dilated lattices within the unit cube.

Contribution

It establishes a uniform bound on the volume of boxes avoiding admissible lattices and determines the optimal dispersion order for dilated lattices.

Findings

Volume of non-intersecting boxes is uniformly bounded.

Dispersion of dilated lattices is of order N^{-1}.

Result is independently obtained by V.N. Temlyakov.

Abstract

In this note we show that the volume of axis-parallel boxes in $\mathbb{R}^d$ which do not intersect an admissible lattice $\mathbb{L}\subset\mathbb{R}^d$ is uniformly bounded. In particular, this implies that the dispersion of the dilated lattices $N^{-1/d}\mathbb{L}$ restricted to the unit cube is of the (optimal) order $N^{-1}$ as $N$ goes to infinity. This result was obtained independently by V.N. Temlyakov (arXiv:1709.08158).