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 ℝⁿ, n ≥ 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 latticepackings, 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, 𝔸^(r) _n , and MW _n, and of the Delone constants of the BCH packings, when n goes to infinity.

Keywords:

• Primary 52C17, 52C23
• Delone set
• sphere packing
• density
• hole