Complexity and Polishability of 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 a sequence of integers $(a_n:n\in \omega )$ such that $$H=\left\{x\in \mathbb{T}:\mathcal{I}\text{-}\underset{n\to \infty }{\lim }{a}_{n}x=0\right\}.$$ We investigate the descriptive complexity and Polishability of these subgroups in terms of the structural and topological properties of the ideal $\mathcal{I}$. Our main structural result shows that $\mathcal{I}$ is an analytic $P$-ideal if and only if all $\mathcal{I}$-characterized subgroups are Polishable. In such case, we explicitly describe a compatible finer Polish group topology. Using results on Polishable subgroups, we obtain a trichotomy for their possible Borel complexities. If $\mathcal{I}$ is a generalized density ideal, we show the sharper dichotomy that every proper $\mathcal{I}$-characterized subgroup is either countable or $F_{\sigma \delta }$-complete. We also prove that this fails for general analytic $P$-ideals by constructing a subgroup characterized by a summable ideal which is neither $F_{\sigma }$ nor $F_{\sigma \delta }$-complete. Finally, we give explicit descriptions of the subgroups associated with the sequences of powers, the Fibonacci sequence, and the sequence of factorials. We conclude with several open questions.