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).

4 pages

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Cited by in corpus (6)

• A remark on the minimal dispersion
• The minimal $k$-dispersion of point sets in high-dimensions
• A tight lower bound on the minimal dispersion
• On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points
• Lower bounds on the minimal dispersion of point sets via cover-free families
• On the fixed volume discrepancy of the Korobov point sets