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

TL;DR

This paper addresses when a statistically characterized subgroup of the circle group $\mathbb{T}$ coincides with the corresponding characterized subgroup. It provides a concise, self-contained proof framework for the structure of $t_{\mathbf{u}}(\mathbb{T})$ and, in particular, shows that for the special sequence $\mathbf{d}$ defined by $(\sharp)$ one has $t_{\mathbf{d}}(\mathbb{T}) = \varphi (\langle \{1/a_n : n \in \mathbb{N}\} \rangle)$ under bounded growth, unifying previous bounded-growth and $\mathbf{d}$-cases. The authors also prove that the statistically characterized subgroup $t_{\mathbf{d}}^s(\mathbb{T})$ equals $t_{\mathbf{d}}(\mathbb{T})$ under natural density conditions (C1) or (C2), providing short proofs of results previously obtained by ADG and DG24. These results deepen the understanding of when statistical convergence yields the same subgroup as ordinary convergence, offering explicit descriptions in terms of generated fractions $1/a_n$.

Abstract

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.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 3.1