Characterizations of groups generated by Kronecker sets
András Biró1
1 A. Rényi Institute of Mathematics Hungarian Academy of Sciences 1053 Budapest, Reáltanoda u. 13-15., Hungary
Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 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 K⊆T be a Kronecker set (a compact set on which every continuous function f:K→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 Z2 (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 K⊆T un ensemble de Kronecker (un ensemble compact sur lequel toute fonction continue f:K→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 Z2 (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].

DOI : 10.5802/jtnb.603

András Biró 1

1 A. Rényi Institute of Mathematics Hungarian Academy of Sciences 1053 Budapest, Reáltanoda u. 13-15., Hungary 
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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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