On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings

Abstract

We study lower bounds of the packing density of a system of nonoverlapping equal spheres in $\rb^{n}, n \geq 2,$ as a function of the maximal circumradius of its Voronoi cells. Our viewpoint, using Delone sets, allows us to investigate the gap between the upper bounds of Rogers or Kabatjanskii-Levenstein and the Minkowski-Hlawka type lower bounds for the density of lattice-packings, without entering the fundamental problem of constructing Delone sets with Delone constants between $2^{-0.401}$ and $1$. As a consequence we provide explicit asymptotic lower bounds of the covering radii (holes) of the Barnes-Wall, Craig, and Mordell-Weil lattices, respectively $BW_{n},$ $\ab_{n}^{(r)},$ and $MW_{n}$, and of the Delone constants of the BCH packings, when $n$ goes to infinity.