Next: Quasi-periodic sphere-packings Up: Maximizing the packing density Previous: Introduction


Sphere packings and dynamical systems

A r-sphere packing S is a countable set of points in tex2html_wrap_inline555 such that the minimal distance between two points is tex2html_wrap_inline615 . Let tex2html_wrap_inline573 be the set of all sphere packings. Define a metric on tex2html_wrap_inline573 by tex2html_wrap_inline621 , where tex2html_wrap_inline623 is the smallest number such that tex2html_wrap_inline625 on the ball tex2html_wrap_inline627 for some tex2html_wrap_inline629 . With this metric, which is adapted from the metric for tilings, tex2html_wrap_inline631 is a complete metric space. Let tex2html_wrap_inline555 act on tex2html_wrap_inline573 by translations tex2html_wrap_inline637 . Given a specific sphere packing S, we can look at the closure tex2html_wrap_inline641 of the orbit tex2html_wrap_inline643 . If tex2html_wrap_inline645 compact in tex2html_wrap_inline573 , we get a dynamical system tex2html_wrap_inline649 , where tex2html_wrap_inline641 is a compact metric space on which tex2html_wrap_inline555 acts by homeomorphisms. Such a system is called minimal or almost periodic, if for every tex2html_wrap_inline655 , the orbit is dense in tex2html_wrap_inline641 , it is called uniquely ergodic, if there exists exactly one tex2html_wrap_inline555 - invariant measure on tex2html_wrap_inline641 , strictly ergodic, if it is uniquely ergodic and minimal.

Remarks.
1) A metric on tex2html_wrap_inline573 , with respect to which the subset of r-packings in tex2html_wrap_inline573 becomes compact is described in [5]. For our purposes, the stronger metric d is good enough.
2) In the statistical mechanics literature, a sphere packing is also called a configuration with hard core restriction.
3) By normalization of the distance in tex2html_wrap_inline555 , we could assume that a sphere packing has radius r=1. We keep the additional parameter r since we are interested in packings tex2html_wrap_inline577 .

We call a sphere packing S rational, if all points of S belong to a d-dimensional lattice tex2html_wrap_inline685 , where U is an invertible tex2html_wrap_inline689 -matrix. On a rational sphere packing, there is a natural tex2html_wrap_inline587 -action, by identifying tex2html_wrap_inline587 with the maximal subgroup of tex2html_wrap_inline555 which leaves the lattice invariant. The packing S consists then of a subset of the lattice and every rational sphere packing defines so a subshift tex2html_wrap_inline699 . This set tex2html_wrap_inline701 is is invariant under the tex2html_wrap_inline587 -action. A rational sphere packings is called strictly ergodic, if the dynamical system tex2html_wrap_inline705 is strictly ergodic. If every shift of the tex2html_wrap_inline587 -action is periodic, S is called periodic,

Remarks.
1) Clearly, a periodic packing is rational and also strictly ergodic.
2) Periodic packings are dense in tex2html_wrap_inline573 since we can periodically continue a given packing outside a given box. Periodic packings are also dense in the set of rational packings.
3) The name almost periodic which stands as a synonym for minimal has no relation with the usual almost periodicity of functions or sequences. The expression almost periodic is however widely used in the topological dynamics and mathematical physics literature.

The lower and upper densities of a sphere packing S are defined as

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where tex2html_wrap_inline717 . If tex2html_wrap_inline719 , then tex2html_wrap_inline721 is called the density of S.

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Remarks.
1) The above lemma is well known and there are other proofs using more theory. The result follows for example also from a multi-dimensional version of Birkhoff's ergodic theorem (see [4] Chapter VIII). The proof given here uses only lightest tools.
2) There exists a dense set of rational packings in tex2html_wrap_inline573 which have no density. Proof. Consider a periodic packing S of radius r having density tex2html_wrap_inline721 . Take an other periodic packing S' of radius r which has density tex2html_wrap_inline769 . Take a first cubic box tex2html_wrap_inline771 centered at zero and fill it with spheres of radius r according to the first packing. Take a second larger cubic box tex2html_wrap_inline775 and fill tex2html_wrap_inline777 with spheres according to the packing in S'. Make tex2html_wrap_inline775 so large that the density of the packing in this finite box tex2html_wrap_inline775 is smaller than tex2html_wrap_inline785 . Take a box tex2html_wrap_inline787 and fill tex2html_wrap_inline789 with spheres according to tex2html_wrap_inline791 make tex2html_wrap_inline787 so large that the density in the box tex2html_wrap_inline787 is larger than tex2html_wrap_inline797 . Continue inductively so that the finite volume densities are alternatively below tex2html_wrap_inline785 and above tex2html_wrap_inline797 .
Given tex2html_wrap_inline803 , we can make tex2html_wrap_inline771 so large that the distance between the original packing and the modified packing is smaller than tex2html_wrap_inline623 .
3) We are forced to define the packing problem on a subclass of packings since the density is not a continuous function on the set of all packings for which the density exists: take such a packing S and define a sequence of packings tex2html_wrap_inline811 obtained from S by deleting all balls in distance less than n from the origin. The packings tex2html_wrap_inline811 have all the same density but tex2html_wrap_inline811 converges to the tex2html_wrap_inline821 , which is a packing with zero density.


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Next: Quasi-periodic sphere-packings Up: Maximizing the packing density Previous: Introduction

Oliver Knill
Mon Jun 22 17:57:55 CDT 1998