
Remarks.
1) A metric on , with respect to
which the subset of rpackings in 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 , we could assume that a
sphere packing has radius r=1.
We keep the additional parameter r since
we are interested in packings .
We call a sphere packing S rational, if all points of S belong to a ddimensional lattice , where U is an invertible matrix. On a rational sphere packing, there is a natural action, by identifying with the maximal subgroup of 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 . This set is is invariant under the action. A rational sphere packings is called strictly ergodic, if the dynamical system is strictly ergodic. If every shift of the 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 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
where . If , then is called the density of S.
Remarks.
1) The above lemma is well known and there are other proofs using more
theory. The result
follows for example also from a multidimensional 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 which have no density.
Proof. Consider
a periodic packing S of radius r having
density . Take an other periodic packing S'
of radius r which has density .
Take a first cubic box centered
at zero and fill it with spheres of radius r according to the first packing.
Take a second larger cubic box and fill
with spheres according to the packing in
S'. Make so large that the density of the packing in this
finite box is smaller than .
Take a box and fill with spheres according to
make so large that the density in the box is larger than
. Continue inductively
so that the finite volume densities are alternatively below
and above
.
Given , we can make so large that the distance
between the original packing and the modified packing is smaller than
.
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 obtained from S
by deleting all balls in distance less than n from the origin.
The packings have all the same density but converges to
the , which is a packing with zero density.