Characterizations of groups generated by Kronecker sets
Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 3, p. 567-582

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 KT be a Kronecker set (a compact set on which every continuous function f:KT 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].

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 KT un ensemble de Kronecker (un ensemble compact sur lequel toute fonction continue f:KT 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].

@article{JTNB_2007__19_3_567_0,
     author = {Bir\'o, Andr\'as},
     title = {Characterizations of groups generated by Kronecker sets},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {3},
     year = {2007},
     pages = {567-582},
     doi = {10.5802/jtnb.603},
     mrnumber = {2388789},
     zbl = {1159.11022},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2007__19_3_567_0}
}
Biró, András. Characterizations of groups generated by Kronecker sets. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 567-582. doi : 10.5802/jtnb.603. http://www.numdam.org/item/JTNB_2007__19_3_567_0/

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