A note on the dispersion of admissible lattices
Provides a theoretical guarantee that admissible lattices are optimal for dispersion, which is important for researchers in discrepancy theory and numerical analysis.
The paper proves that the dispersion of dilated admissible lattices in the unit cube achieves the optimal order N^{-1}, showing that such lattices are optimal point sets for numerical integration and approximation.
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).