Maximizing the Packing Density on a Class of Almost Periodic Sphere Packings

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 Rd defines a dynamical system with time Rd. If the dynam ical system is strictly ergodic, the packing has a well defined density. The packings considered here belong to quasi-periodic dynamical systems, strictly ergodic transla tions 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 spherepacking is a periodic or aperiodic Voronoi tiling of Rd 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. 0723-0869/96/030227-246$5.50 2 2 8 O l i v e r K n i l l