Characterizations of groups generated by Kronecker sets

Biró, András

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

Ces dernières années, depuis l’article [B-D-S], nous avons étudié la possibilité de caratériser les sous-groupes dénombrables du tore $T=\mathbf{R}/\mathbf{Z}$ par des sous-ensembles de $\mathbf{Z}$. 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 ${\mathbf{Z}}^2$ (au lieu d’un sous-ensemble de $\mathbf{Z}$). 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 $\mathbf{Z}$ (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 $T=\mathbf{R}/\mathbf{Z}$ by subsets of $\mathbf{Z}$. 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 ${\mathbf{Z}}^2$ (instead of a subset of $\mathbf{Z}$). 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 $\mathbf{Z}$ (or an integer sequence) in the sense of [B-D-S].

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