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

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.

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.

Published online:
DOI: 10.5802/jtnb.489
Gilbert Muraz 1; Jean-Louis Verger-Gaugry 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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Gilbert Muraz; Jean-Louis Verger-Gaugry. On a generalization of the Selection Theorem of Mahler. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 237-269. doi : 10.5802/jtnb.489. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.489/

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