Element-wise description of the $\mathcal{I}$-characterized subgroups of the circle

Abstract

According to Cartan, given an ideal $\mathcal{I}$ of $\mathbb{N}$, a sequence $(x_n{)}_{n\in \mathbb{N}}$ in the circle group $\mathbb{T}$ is said to {\em $\mathcal{I}$-converge} to a point $x\in \mathbb{T}$ if $\{n\in \mathbb{N}:x_n\notin U\}\in \mathcal{I}$ for every neighborhood $U$ of $x$ in $\mathbb{T}$. For a sequence $\mathbf{u}=(u_n{)}_{n\in \mathbb{N}}$ in $\mathbb{Z}$, let $$t_{\mathbf{u}}^{\mathcal{I}}(\mathbb{T}):=\{x\in \mathbb{T}:u_{n}x\text{I-converges to}0\}.$$ For an analytic free $P$-ideal $\mathcal{I}$, this set is a Borel (hence, Polishable) subgroup of $\mathbb{T}$ with many nice properties, largely studied in the case when $\mathcal{I}=\mathcal{F}in$ is the ideal of all finite subsets of $\mathbb{N}$ (so $\mathcal{F}in$-convergence coincides with the usual one) for its remarkable connection to topological algebra, descriptive set theory and harmonic analysis. We give a complete element-wise description of $t_{\mathbf{u}}^{\mathcal{I}}(\mathbb{T})$ when $\mathbf{u}$ is a strictly increasing sequence of positive integers with $u_0=1$ and $u_n\mid u_{n+1}$ for every $n\in \mathbb{N}$ and under suitable hypotheses on $\mathcal{I}$. In the special case when $\mathcal{I}=\mathcal{F}in$, we obtain an alternative proof of a simplified version of a known result.