Characterized subgroups on the unit circle

Abstract

Given an ideal $\mathcal{I}$ on $\omega$, a subgroup $H$ of the unit circle $\mathbb{T}$ is said to be $\mathcal{I}$-characterized if there exists an integer sequence $a=(a_n: n \in \omega)$ such that $$ H=\mathsf{H}_{a}(\mathcal{I}):= \left\{x\in\mathbb{T}:\mathcal I\text{-}\lim_{n\to \infty} a_nx=0\right\}. $$ We also consider the corresponding $\mathcal{I}^\star$-version. We provide upper bounds for the topological complexities of those subgroups in terms of the complexity of $\mathcal{I}$. Moreover, we prove that Rudin--Keisler and Rudin--Blass reductions between ideals induce inclusions between the corresponding families of characterized subgroups. As a consequence, every characterized subgroup, and in particular every countable subgroup of $\mathbb{T}$, is $\mathcal{I}$-characterized for every meager ideal $\mathcal{I}$. We also show that if the image of $(a_n: n \in \omega)$ contains arbitrarily large intervals, then every subgroup of $\mathbb{T}$ can be written as $\mathsf{H}_{a}(\mathcal{J})$ for some ideal $\mathcal{J}=\mathcal{J}_{H,a}$. We analyze the descriptive complexity and $P$-properties of these ideals. Finally, we study when the equality $\mathsf H_{a}(\mathcal{I})=\mathbb{T}$ forces $\mathrm{supp}(a)\in\mathcal{I}$. We prove this for a class of ideals satisfying a Katetov-type condition involving $\mathcal{ED}$, including nowhere tall ideals as well as the ideals $\mathsf{nwd}$ and $\mathsf{null}$. We also obtain non-inclusion results between families of $\mathcal{I}$-characterized subgroups: for instance, we show that if the ideal $\mathcal{I}$ is tall and translation invariant then the subgroup $\mathsf{H}_{(2^n)}(\mathcal{I})$ cannot be characterized. We use our results to answer several open problems posed in the literature.