On Self-Similar Finitely Generated Uniformly Discrete (SFU-) sets and Sphere Packings

Jean-Louis Verger-Gaugry (1)

1 IF - Institut Fourier

This 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 $\Lambda$ in ~$\rb^{n}$ 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 $\zb$-rank of $\zb[\Lambda-\Lambda]$. 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.

Mots clés: sphere packing; Delone set; Meyer set; mathematical quasicrystal; self-similarity; algebraic integer; Euclidean minimum; inhomogeneous minimum; hole; density; sphere packing.

Domaines: Théorie des nombres [math.NT]

Jean-Louis Verger-Gaugry. On Self-Similar Finitely Generated Uniformly Discrete (SFU-) sets and Sphere Packings. IRMA - Lectures in Mathematics and Theoretical Physics, 2006, 10, pp.39--78. ⟨hal-00352786⟩