Characterizations of groups generated by Kronecker sets

András Biró[1]

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

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 3, page 567-582
  • ISSN: 1246-7405

Abstract

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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 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].

How to cite

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Biró, András. "Characterizations of groups generated by Kronecker sets." Journal de Théorie des Nombres de Bordeaux 19.3 (2007): 567-582. <http://eudml.org/doc/249960>.

@article{Biró2007,
abstract = {In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T=\{\bf R\}/\{\bf Z\}$ by subsets of $\{\bf Z\}$. Here we consider new types of subgroups: let $K\subseteq T$ be a Kronecker set (a compact set on which every continuous function $f:K\rightarrow T$ 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 $\{\bf Z\}^2$ (instead of a subset of $\{\bf Z\}$). 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 $\{\bf Z\}$ (or an integer sequence) in the sense of [B-D-S].},
affiliation = {A. Rényi Institute of Mathematics Hungarian Academy of Sciences 1053 Budapest, Reáltanoda u. 13-15., Hungary},
author = {Biró, András},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {torus approximation; Kronecker sets},
language = {eng},
number = {3},
pages = {567-582},
publisher = {Université Bordeaux 1},
title = {Characterizations of groups generated by Kronecker sets},
url = {http://eudml.org/doc/249960},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Biró, András
TI - Characterizations of groups generated by Kronecker sets
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 3
SP - 567
EP - 582
AB - In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T={\bf R}/{\bf Z}$ by subsets of ${\bf Z}$. Here we consider new types of subgroups: let $K\subseteq T$ be a Kronecker set (a compact set on which every continuous function $f:K\rightarrow T$ 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 ${\bf Z}^2$ (instead of a subset of ${\bf Z}$). 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 ${\bf Z}$ (or an integer sequence) in the sense of [B-D-S].
LA - eng
KW - torus approximation; Kronecker sets
UR - http://eudml.org/doc/249960
ER -

References

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