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$$\mathbb{T}$0^H = {(z_n)\in $\mathbb{T}$^{\infty} : z_n\to 1 }$\mathrm{.}Let$G$beasubgroupof$X$\mathrm{.}Weprovethat$G=s{$\mathbf{u}$} (X)$forsome$$\mathbf{u}$$iffitcanberepresentedassomeduallyclosedsubgroup$G_{$\mathbf{u}$}$of${\rm Cl}X G \times $\mathbb{T}$0^H$\mathrm{.}Inparticular,$s{$\mathbf{u}$} (X)$ispolishable\mathrm{.}Let$$\mathbf{u}$={u_n}$bea$T$-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.