On a generalization of the Selection Theorem of Mahler
Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 237-269.

On montre que l’ensemble 𝒰𝒟 r des ensembles de points de n ,n1, qui ont la propriété que leur distance interpoint minimale est plus grande qu’une constante strictement positive r>0 donnée est muni d’une métrique pour lequel il est compact et tel que la métrique de Hausdorff sur le sous-ensemble 𝒰𝒟 r,f 𝒰𝒟 r des ensembles de points finis est compatible avec la restriction de cette topologie à 𝒰𝒟 r,f . Nous montrons que ses ensembles de Delaunay (Delone) de constantes données dans n ,n1, sont compacts. Trois (classes de) métriques, dont l’une de nature cristallographique, nécessitant un point base dans l’espace ambiant, sont données avec leurs propriétés, pour lesquelles nous montrons qu’elles sont topologiquement équivalentes. On prouve que le processus d’enlèvement de points est uniformément continu à l’infini. Nous montrons que ce Théorème de compacité implique le Théorème classique de Sélection de Mahler. Nous discutons la généralisation de ce résultat à des espaces ambiants autres que n . L’espace 𝒰𝒟 r est l’espace des empilements de sphères égales de rayon r/2.

The set 𝒰𝒟 r of point sets of n ,n1, having the property that their minimal interpoint distance is greater than a given strictly positive constant r>0 is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset 𝒰𝒟 r,f 𝒰𝒟 r of the finite point sets is compatible with the restriction of this topology to 𝒰𝒟 r,f . We show that its subsets of Delone sets of given constants in n ,n1, are compact. Three (classes of) metrics, whose one of crystallographic nature, requiring a base point in the ambient space, are given with their corresponding properties, for which we show topological equivalence. The point-removal process is proved to be uniformly continuous at infinity. We prove that this compactness Theorem implies the classical Selection Theorem of Mahler. We discuss generalizations of this result to ambient spaces other than n . The space 𝒰𝒟 r is the space of equal sphere packings of radius r/2.

DOI : 10.5802/jtnb.489
Muraz, Gilbert 1 ; Verger-Gaugry, Jean-Louis 1

1 Institut Fourier - CNRS UMR 5582 Université de Grenoble I BP 74 - Domaine Universitaire 38402 Saint Martin d’Hères, France
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Muraz, Gilbert; Verger-Gaugry, Jean-Louis. On a generalization of the Selection Theorem of Mahler. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 237-269. doi : 10.5802/jtnb.489. http://www.numdam.org/articles/10.5802/jtnb.489/

[BL] M. Baake, D. Lenz, Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergod. Th. & Dynam. Sys. 24 (6) (2004), 1867–1893. | MR | Zbl

[B] N. Bourbaki, General Topology. Chapter IX, Addison-Wesley, Reading, 1966.

[Bo] L. Bowen, On the existence of completely saturated packings and completely reduced coverings. Geom. Dedicata 98 (2003), 211–226. | MR | Zbl

[Ca] J.W.S. Cassels, An introduction to the Geometry of Numbers. Springer Verlag, 1959. | Zbl

[Ch] C. Chabauty, Limite d’Ensembles et Géométrie des Nombres. Bull. Soc. Math. Fr. 78 (1950), 143–151. | EuDML | Numdam | MR | Zbl

[CS] J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups. Springer-Verlag, Berlin, 1999, third edition. | MR | Zbl

[Dw] S. Dworkin, Spectral Theory and X-Ray Diffraction. J. Math. Phys. 34 (7) (1993), 2965–2967. | MR | Zbl

[GVG] J.-P. Gazeau, J.-L. Verger-Gaugry, Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. J. Théorie Nombres Bordeaux 16 (2004), 125–149. | EuDML | Numdam | MR | Zbl

[Go] J.-B. Gouéré, Quasicrystals and almost-periodicity. Comm. Math. Phys. (2005), accepted. | MR | Zbl

[Groe] H. Groemer, Continuity properties of Voronoi domains. Monatsh. Math. 75 (1971), 423–431. | MR | Zbl

[Gr] E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985. | MR | Zbl

[GL] P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers. North-Holland, 1987. | MR | Zbl

[Ha] T.C. Hales, Sphere Packings I. Discrete Comput. Geom. 17 (1997), 1–51. | MR | Zbl

[Ke] J.L. Kelley, Hyperspaces of a Continuum. Trans. Amer. Math. Soc. 52 (1942), 22–36. | MR | Zbl

[La] J.C. Lagarias, Bounds for Local Density of Sphere Packings and the Kepler Conjecture. Discrete Comput. Geom. 27 (2002), 165–193. | MR | Zbl

[Ma] K. Mahler, On Lattice Points in n-dimensional Star Bodies.I. Existence Theorems. Proc. Roy. Soc. London A 187 (1946), 151–187. | MR | Zbl

[MS] A.M. Macbeath, S. Swierczkowski, Limits of lattices in a compactly generated group. Canad. J. Math. 12 (1960), 427-437. | MR | Zbl

[Mf] R.B. Mcfeat, Geometry of numbers in adele spaces. Dissertationes Math. (Rozprawy mat.), Warsawa, 88 (1971), 1–49. | MR | Zbl

[Mi] E. Michael, Topologies on Spaces of Subsets. Trans. Amer. Math. Soc. 71 (1951), 152–182. | MR | Zbl

[Mo] R.V. Moody, Meyer sets and their duals. In The Mathematics of Long-Range Aperiodic Order, Ed. by R.V. Moody, Kluwer Academic Publishers (1997), 403–441. | MR | Zbl

[Mu] D. Mumford, A Remark on Mahler’s Compactness Theorem. Proc. of the Amer. Math. Soc. 28 (1971), 289–294. | MR | Zbl

[MVG] G. Muraz, J.-L. Verger-Gaugry, On lower bounds of the density of Delone sets and holes in sequences of sphere packings. Exp. Math. 14:1 (2005), 49–59. | MR | Zbl

[MVG1] G. Muraz, J.-L. Verger-Gaugry, On continuity properties of Voronoi domains and a theorem of Groemer. Preprint (2004).

[RW] C. Radin, M. Wolff, Space Tilings and Local Isomorphism. Geom. Dedicata 42 (1992), 355–360. | MR | Zbl

[Rob] E. Arthur Robinson, Jr., The Dynamical Theory of Tilings and Quasicrystallography. In Ergodic Theory of d -actions, (Warwick 1993-4), London Math. Soc. Lec. Note Ser. 228, Cambridge Univ. Press, Cambridge, 451–473. | MR | Zbl

[Ro] C.A. Rogers, Packing and Covering. Cambridge University Press, 1964. | MR | Zbl

[RSD] K. Rogers, H.P.F. Swinnerton-Dyer, The Geometry of Numbers over algebraic number fields. Trans. Amer. Math. Soc. 88 (1958), 227–242. | MR | Zbl

[So] B. Solomyak, Spectrum of Dynamical Systems Arising from Delone Sets. In Quasicrystals and Discrete Geometry, Fields Institute Monographs, J. Patera Ed., 10 (1998), AMS, 265–275. | MR | Zbl

[St] K.B. Stolarsky, Sums of distances between points on a sphere. Proc. Amer. Math. Soc. 35 (1972), 547–549. | MR | Zbl

[We] A. Weil, Sur les Espaces à Structure Uniforme et sur la Topologie Générale. Hermann, Paris, 1938. | Zbl

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