Experimental Mathematics

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

G. Muraz and J. -L. Verger-Gaugry

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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.

Article information

Experiment. Math., Volume 14, Issue 1 (2005), 47-57.

First available in Project Euclid: 30 June 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31] 52C23: Quasicrystals, aperiodic tilings

Delone set sphere packing density hole


Muraz, G.; Verger-Gaugry, J. -L. On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings. Experiment. Math. 14 (2005), no. 1, 47--57. https://projecteuclid.org/euclid.em/1120145569

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