Characterizations of groups generated by Kronecker sets

András Biró

doi:10.5802/jtnb.603

András Biró

Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 567-582.

Résumé
Abstract

Ces dernières années, depuis l’article [B-D-S], nous avons étudié la possibilité de caratériser les sous-groupes dénombrables du tore $T = R / Z$ par des sous-ensembles de $Z$ . Nous considérons ici de nouveaux types de sous-groupes : soit $K \subseteq T$ un ensemble de Kronecker (un ensemble compact sur lequel toute fonction continue $f : K \to T$ peut être approchée uniformément par des caractéres de $T$ ) et $G$ le groupe engendré par $K$ . Nous prouvons (théorème 1) que $G$ peut être caractérisé par un sous-ensemble de $Z^{2}$ (au lieu d’un sous-ensemble de $Z$ ). Si $K$ est fini, le théorème 1 implique notre résultat antérieur de [B-S]. Nous montrons également (théorème 2) que si $K$ est dénombrable alors $G$ ne peut pas être caractérisé par un sous-ensemble de $Z$ (ou une suite d’entiers) au sens de [B-D-S].

In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T = R / Z$ by subsets of $Z$ . Here we consider new types of subgroups: let $K \subseteq T$ be a Kronecker set (a compact set on which every continuous function $f : K \to T$ can be uniformly approximated by characters of $T$ ), and $G$ the group generated by $K$ . We prove (Theorem 1) that $G$ can be characterized by a subset of $Z^{2}$ (instead of a subset of $Z$ ). If $K$ is finite, Theorem 1 implies our earlier result in [B-S]. We also prove (Theorem 2) that if $K$ is uncountable, then $G$ cannot be characterized by a subset of $Z$ (or an integer sequence) in the sense of [B-D-S].

Reçu le : 2005-05-12
Publié le : 2010-09-26

MR 2388789 | Zbl 1159.11022

DOI : https://doi.org/10.5802/jtnb.603

@article{JTNB_2007__19_3_567_0,
     author = {Andr\'as Bir\'o},
     title = {Characterizations of groups generated by {Kronecker} sets},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {567--582},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {3},
     year = {2007},
     doi = {10.5802/jtnb.603},
     zbl = {1159.11022},
     mrnumber = {2388789},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.603/}
}

András Biró. Characterizations of groups generated by Kronecker sets. Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 567-582. doi : 10.5802/jtnb.603. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.603/

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