On $T$-sequences and characterized subgroups

TL;DR

This paper characterizes subgroups of compact metrizable abelian groups via $T$-sequences, showing their structure, polishability, and duality properties, and discusses limitations in characterizing groups generated by Kronecker sets.

Contribution

It provides a complete characterization of subgroups as $s_{f u}(X)$ using $T$-sequences and explores their duality and topological properties, including polishability.

Findings

Subgroups $s_{f u}(X)$ are characterized by dually closed subgroups.

$s_{f u}(X)$ is shown to be polishable.

Groups generated by Kronecker sets cannot be characterized.

Abstract

Let $X$ be a compact metrizable abelian group and $\mathbf{u}=\{u_n\}$ be a sequence in its dual $X^{\wedge}$. Set $s_{\mathbf{u}} (X)= \{x: (u_n,x)\to 1\}$ and $\mathbb{T}_0^H = \{(z_n)\in \mathbb{T}^{\infty} : z_n\to 1 \}$. Let $G$ be a subgroup of $X$. We prove that $G=s_{\mathbf{u}} (X)$ for some $\mathbf{u}$ iff it can be represented as some dually closed subgroup $G_{\mathbf{u}}$ of ${\rm Cl}_X G \times \mathbb{T}_0^H$. In particular, $s_{\mathbf{u}} (X)$ is polishable. Let $\mathbf{u}=\{u_n\}$ be a $T$-sequence. Denote by $(\widehat{X}, \mathbf{u})$ the group $X^{\wedge}$ equipped with the finest group topology in which $u_n \to 0$. It is proved that $(\widehat{X}, \mathbf{u})^{\wedge} =G_{\mathbf{u}}$ and $\mathbf{n} (\widehat{X}, \mathbf{u}) = s_{\mathbf{u}} (X)^{\perp}$. We also prove that the group generated by a Kronecker set can not be characterized.