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 $\mathrm{supp}(r):=\{n\in\mathbb N_+: c_n\neq0\}$ 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 $u_nx$ $\mathcal I$-converges to $0$, that is, $\{n\in \mathbb N: u_nx \not \in U\}\in \mathcal I$ 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 $\mathcal I$.