On self-similar finitely generated uniformly discrete (SFU-)sets and sphere packings

Abstract

The first part of this paper is a survey on links between geometry of numbers and aperiodic crystals in physics, viewed from the mathematical side. In a second part, we prove the existence of a canonical cut-and-project scheme above a (SFU-set) self-similar finitely generated packing of (equal) spheres Λ in ℝn and investigate its consequences, in particular the role played by the Euclidean and inhomogeneous minima of the algebraic number field generated by the self-similarity on the Delone constant of the sphere packing. We discuss the isolation phenomenon. The degree d of this field divides the ℤ-rank of ℤ[Λ − Λ]. We give a lower bound of the Delone constant of a k-thin SFU-set (sphere packing) which arises from a model set or a Meyer set when d is large enough.