×

On self-similar finitely generated uniformly discrete (SFU-)sets and sphere packings. (English) Zbl 1170.52303

Nyssen, Louise (ed.), Physics and number theory. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-028-0/pbk). IRMA Lectures in Mathematics and Theoretical Physics 10, 39-78 (2006).
This paper, written for the 75ème Rencontres between Mathematicians and Physicists held at IRMA, Strasbourg, describes the connection between the Geometry of Numbers and aperiodic crystals in Physics from the point of view of the mathematics. The author defines uniformly discrete sets and Delone sets in both a metric space which is \(\sigma\)-compact and locally compact and when the ambient space is \(\mathbb R^n\) with a cut-and-project scheme that lies above it with a locally compact abelian group as internal space. Then a classification of discrete sets and Delone sets in \(\mathbb R^n\) along with their relative inclusions are given. The author proves the existence of a canonical cut-and-project scheme above a self-similar finitely generated sphere packing in \(\mathbb R^n\). Consequences of this theorem are discussed, including 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. Two origins of (pseudo)-Delone constants of sphere packings are given, the first from the geometrical properties of the central cluster and the second from a purely arithmetic standpoint. Lower bounds are given for densities and the pseudo-Delone constant and on the Delone constant of a self-similar finitely generated sphere packing. An extensive bibliography of the relevant literature is given.
For the entire collection see [Zbl 1098.11003].

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
11H06 Lattices and convex bodies (number-theoretic aspects)
PDFBibTeX XMLCite