Nested ideals and topologically ${\mathbf{u}}_{\mathcal{I}}$-torsion elements of the circle group

Abstract

Let $\mathbf{u}=(u_n{)}_{n\in \mathbb{N}}$ be a sequence in ${\mathbb{N}}_{+}$ with $u_0=1$ and $u_n\mid u_{n+1}$ for every $n\in \mathbb{N}$, and let $b_n:=u_{n+1}/u_n$ for every $n\in {\mathbb{N}}_{+}$. For every $r\in [0,1)$, there exists a unique sequence $(c_n{)}_{n\in {\mathbb{N}}_{+}}$ in $\mathbb{N}$ such that $r={\sum }_{n=1}^{\infty }\frac{c_n}{u_n}$, with $c_n<b_n$ for every $n\in {\mathbb{N}}_{+}$, and $c_n<b_n-1$ for infinitely many $n\in {\mathbb{N}}_{+}$; let $\mathrm{supp}(r):=\{n\in {\mathbb{N}}_{+}:c_n\ne 0\}$ and ${\mathrm{supp}}_b(r):=\{n\in {\mathbb{N}}_{+}:c_n=b_n-1\}$. For $x=r+\mathbb{Z}\in \mathbb{T}$, let $\mathrm{supp}(x)=\mathrm{supp}(r)$ and ${\mathrm{supp}}_b(x)={\mathrm{supp}}_b(r)$. For an ideal $\mathcal{I}$ of $\mathbb{N}$, an element $x$ of the circle group $\mathbb{T}$ is called a topologically ${\mathbf{u}}_{\mathcal{I}}$-torsion element of $\mathbb{T}$ if $u_{n}x$ $\mathcal{I}$-converges to $0$, that is, $\{n\in \mathbb{N}:u_{n}x\notin U\}\in \mathcal{I}$ for every neighborhood $U$ of $0$ in $\mathbb{T}$. In this paper, under suitable conditions on the ideal $\mathcal{I}$, we completely describe the ${\mathbf{u}}_{\mathcal{I}}$-torsion elements $x$ of $\mathbb{T}$ with ${\lim }_{n\in \mathrm{supp}(x)}{b}_n=\infty$ and those with $\{b_n:n\in \mathrm{supp}(x)\}$ bounded. According to Corollary 2.12 in [A. Ghosh, Ric. Mat. 73 (2024), 2263--2281], an element $x\in \mathbb{T}$ with $\{b_n:n\in \mathrm{supp}(x)\}$ bounded is topologically ${\mathbf{u}}_{\mathcal{I}}$-torsion if and only if $\mathrm{supp}(x)+1\setminus \mathrm{supp}(x)\in \mathcal{I}$ and $\mathrm{supp}(x)\setminus {\mathrm{supp}}_b(x)\in \mathcal{I}$. We characterize the ideals $\mathcal{I}$ of $\mathbb{N}$, naming them nested, such that this equivalence holds and we provide examples of non-nested ideals $\mathcal{I}$ that satisfy the above mentioned suitable conditions, so that the equivalence claimed by Ghosh fails for those $\mathcal{I}$.