Knill O.
Maximizing the packing density on a class of
       almost periodic sphere packings
(50K, LaTeX)

ABSTRACT.  We consider the variational problem of maximizing the packing density
on some finite dimensional set of almost periodic sphere packings. We
show that the maximal density on this manifold is obtained by periodic
packings. Since the density is a continuous, but a nondifferentiable function
on this manifold, the variational problem is related to
number theoretical questions.
Every sphere packing in $\RR^d$ defines
a dynamical system with time $\RR^d$. If the dynamical
system is strictly ergodic, the packing
has a well defined density. The packings considered here
belong to quasi-periodic dynamical systems, strictly ergodic
translations on a compact topological group and are higher dimensional
versions of circle sequences in one dimension. In most cases,
these packings are quasicrystals because the dynamics has
dense point spectrum.
Attached to each quasi-periodic sphere-packing
is a periodic or aperiodic Voronoi tiling of $\RR^d$ by
finitely many types of polytopes. Most of the tilings
belonging to the $d$-dimensional set of packings are aperiodic.
We construct a one-parameter family of dynamically isospectral
quasi-periodic sphere packings
which have a uniform lower bound on the density when variing the radius
of the packing. The simultaneous density bound depends on constants in
the theory of simultaneous Diophantine approximation.