On a generalization of the Selection Theorem of Mahler

Muraz, Gilbert; Verger-Gaugry, Jean-Louis

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

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 237-269.

Résumé
Abstract

On montre que l’ensemble $𝒰 𝒟_{r}$ des ensembles de points de $ℝ^{n}, n \geq 1$ , 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 $𝒰 𝒟_{r, f} \subset 𝒰 𝒟_{r}$ des ensembles de points finis est compatible avec la restriction de cette topologie à $𝒰 𝒟_{r, f}$ . Nous montrons que ses ensembles de Delaunay (Delone) de constantes données dans $ℝ^{n}, n \geq 1$ , 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 $ℝ^{n}$ . L’espace $𝒰 𝒟_{r}$ est l’espace des empilements de sphères égales de rayon $r / 2$ .

The set $𝒰 𝒟_{r}$ of point sets of $ℝ^{n}, n \geq 1$ , 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 $𝒰 𝒟_{r, f} \subset 𝒰 𝒟_{r}$ of the finite point sets is compatible with the restriction of this topology to $𝒰 𝒟_{r, f}$ . We show that its subsets of Delone sets of given constants in $ℝ^{n}, n \geq 1$ , 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 $ℝ^{n}$ . The space $𝒰 𝒟_{r}$ is the space of equal sphere packings of radius $r / 2$ .

MR Zbl

DOI : 10.5802/jtnb.489

Affiliations des auteurs :

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. https://www.numdam.org/articles/10.5802/jtnb.489/

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