A note on characterized and statistically characterized subgroups of $\mathbb{T}=\mathbb{R}/\mathbb{Z}$

P. Das, A. Ghosh and T. Aziz has given in \cite[Theorem 3.15]{ADG} a result on statistically characterized subgroups of the circle group $\mathbb{R}/\mathbb{Z}$, which answers, together with \cite[Corollary 2.4]{DG24}, questions of \cite{DDB} and \cite{DG25}. Here we give a completely different and much shorter proof of these results.