A note on the dispersion of admissible lattices

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).