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 $\{n\in \mathbb N: x_n \not \in U\}\in \mathcal I$ for every neighborhood $U$ of $x$ in $\mathbb T$. For a sequence $\mathbf u=(u_n)_{n\in\mathbb N}$ in $\mathbb Z$, let $$t_{\mathbf u}^\mathcal I(\mathbb T) :=\{x\in \mathbb T: u_nx \ \text{$\mathcal I$-converges to}\ 0 \}.$$ 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 $u_n\mid u_{n+1}$ 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.