Dense forests and Danzer sets

Solomon, Yaar; Weiss, Barak

doi:10.24033/asens.2303

Dense forests and Danzer sets
[Forêts denses et ensembles de Danzer]

Solomon, Yaar ; Weiss, Barak

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1053-1074.

Résumé
Abstract

Un ensemble de Danzer est une partie $Y$ de $ℝ^{d}$ qui rencontre tout ensemble convexe de volume 1. On ne sait pas s'il existe des ensembles de Danzer dans $ℝ^{d}$ de croissance $O (T^{d})$ . Nous démontrons que les candidats naturels, tels que les ensembles discrets produits à l'aide de substitutions, de sections et de projections, ne sont pas des ensembles de Danzer. Dans le cas des sections et projections, notre preuve repose sur la dynamique et la structure des réseaux dans les groupes algébriques. Nous considérons aussi une notion plus faible, l'existence d'une forêt dense uniformément discrète, et nous utilisons la dynamique homogène (en particulier les théorèmes de Ratner sur les flots unipotents) pour construire de tels ensembles. Nous démontrons aussi l'équivalence entre le problème de Danzer et un problème combinatoire classique et en déduisons l'existence d'ensembles de Danzer de croissance $O (T^{d} log T)$ , améliorant ainsi la borne précédente $O (T^{d} {log}^{d - 1} T)$ .

A set $Y \subseteq ℝ^{d}$ that intersects every convex set of volume 1 is called a Danzer set. It is not known whether there are Danzer sets in $ℝ^{d}$ with growth rate $O (T^{d})$ . We prove that natural candidates, such as discrete sets that arise from substitutions and from cut-and-project constructions, are not Danzer sets. For cut and project sets our proof relies on the dynamics of homogeneous flows. We consider a weakening of the Danzer problem, the existence of a uniformly discrete dense forest, and we use homogeneous dynamics (in particular Ratner's theorems on unipotent flows) to construct such sets. We also prove an equivalence between the above problem and a well-known combinatorial problem, and deduce the existence of Danzer sets with growth rate $O (T^{d} log T)$ , improving the previous bound of $O (T^{d} {log}^{d - 1} T)$ .

Publié le : 2016-11-12

MR Zbl

DOI : 10.24033/asens.2303

Classification : 52C17, 52C23, 37A17.
Keywords: Discrete sets, Danzer problem, substitution tilings, cut and project sets.
Mot clés : Ensembles discrets, problème de Danzer, pavages de substitution, sections et projections.

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     author = {Solomon, Yaar and Weiss, Barak},
     title = {Dense forests and {Danzer} sets},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1053--1074},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {5},
     year = {2016},
     doi = {10.24033/asens.2303},
     mrnumber = {3581810},
     zbl = {1379.52020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2303/}
}

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Solomon, Yaar; Weiss, Barak. Dense forests and Danzer sets. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1053-1074. doi : 10.24033/asens.2303. http://archive.numdam.org/articles/10.24033/asens.2303/

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