Characterized subgroups of the circle group

Abstract

A subgroup H of the circle group $$\mathbb T$$ is said to be characterized by a sequence $$\mathbf {u}= (u_n)_{n\in \mathbb N}$$ of integers if $$H=\{x\in \mathbb T: u_nx\rightarrow 0\}$$ . The characterized subgroups of $$\mathbb T$$ are known also under the name topologically $$\mathbf {u}$$ -torsion subgroups. This survey paper is dedicated to the characterized subgroups of $$\mathbb T$$ : we recall their main properties and collect most of the basic results from the wide bibliography, following, when possible, the historical line, and trying to show the deep roots of this topic in several areas of Mathematics. Due to this universality of the topic, many notions and results were found independently by various authors working unaware of each other, so our effort is also directed towards giving credit to all of them to the best of our knowledge. We provide also some background on the notions of characterized subgroup and topologically $$\mathbf {u}$$ -torsion subgroup in the general case of topological abelian groups, where they differ very substantially.