On $T$-sequences and characterized subgroups

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.

💡 Research Summary

The paper investigates the interplay between compact metrizable abelian groups and sequences of characters, focusing on the class of subgroups that can be described as “characterized” by a sequence. Let (X) be a compact metrizable abelian group and let (\mathbf{u}={u_{n}}\subset X^{\wedge}) be a sequence of continuous characters. The authors define
\