A note on the dispersion of admissible lattices

In this note we show that the volume of axis-parallel boxes in R which do not intersect an admissible lattice L ⊂ R is uniformly bounded. In particular, this implies that the dispersion of the dilated lattices NL restricted to the unit cube is of the (optimal) order N as N goes to infinity. This result was obtained independently by V.N. Temlyakov (arXiv:1709.08158). Let L ⊂ R, d ∈ N, be an admissible lattice (in the sense of Skriganov [17]), i.e., L = T (Z) for some T ∈ Rd×d such that (1) Nm(L) := inf z∈L\{0} d