In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus by subsets of . Here we consider new types of subgroups: let be a Kronecker set (a compact set on which every continuous function can be uniformly approximated by characters of ), and the group generated by . We prove (Theorem 1) that can be characterized by a subset of (instead of a subset of ). If is finite, Theorem 1 implies our earlier result in [B-S]. We also prove (Theorem 2) that if is uncountable, then cannot be characterized by a subset of (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 par des sous-ensembles de . Nous considérons ici de nouveaux types de sous-groupes : soit un ensemble de Kronecker (un ensemble compact sur lequel toute fonction continue peut être approchée uniformément par des caractéres de ) et le groupe engendré par . Nous prouvons (théorème 1) que peut être caractérisé par un sous-ensemble de (au lieu d’un sous-ensemble de ). Si est fini, le théorème 1 implique notre résultat antérieur de [B-S]. Nous montrons également (théorème 2) que si est dénombrable alors ne peut pas être caractérisé par un sous-ensemble de (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}, 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 = {http://archive.numdam.org/articles/10.5802/jtnb.603/} }
TY - JOUR AU - Biró, András TI - Characterizations of groups generated by Kronecker sets JO - Journal de théorie des nombres de Bordeaux PY - 2007 SP - 567 EP - 582 VL - 19 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.603/ DO - 10.5802/jtnb.603 LA - en ID - JTNB_2007__19_3_567_0 ER -
%0 Journal Article %A Biró, András %T Characterizations of groups generated by Kronecker sets %J Journal de théorie des nombres de Bordeaux %D 2007 %P 567-582 %V 19 %N 3 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.603/ %R 10.5802/jtnb.603 %G en %F 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://archive.numdam.org/articles/10.5802/jtnb.603/
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