Anna Giordano Bruno

Ideal convergence and characterized subgroups of the circle group.

(Based on a joint work with: D. Dikranjan, R. Di Santo and H. Weber)

Recently several generalizations were introduced of the notion of characterized subgroup t_{\mathbf u}(\mathbb T):=\{x\in\mathbb T: u_nx\to0\} of the circle group \mathbb T by a sequence \mathbf u=(u_n)_{n\in\mathbb N} of integers. They are based on weaker notions of convergence, among them the most general one is due to Cartan: for an ideal \mathcal I of \mathbb N, a sequence (y_n)_{n\in\mathbb N} in \mathbb T is said to \mathcal I-converge to a point y\in \mathbb T, denoted by y_n\overset{\mathcal I}\to y, if \{n\in\mathbb N: y_n \not \in U\}\in \mathcal I for every neighborhood U of y in \mathbb T. Das and Ghosh defined a subgroup H of \mathbb T to be \mathcal I-characterized with respect to \mathcal I by a sequence of integers \mathbf u = (u_n)_{n\in\mathbb N} if H coincides with t^\mathcal I_\mathbf u(\mathbb T):=\{x\in\mathbb T: u_nx\overset{\mathcal I}\to 0\}. When \mathcal I=\mathcal Fin is the ideal of all finite subsets of \mathbb N, \mathcal Fin-convergence is the usual convergence and t_{\mathbf u}^{\mathcal Fin}(\mathbb T)=t_{\mathbf u}(\mathbb T).

The talk starts with a brief history on classical characterized subgroups of \mathbb T and their relevance in several areas of Mathematics where the behavior of the sequence (u_nx)_{n\in\mathbb N} as above is studied, as Topological Algebra, Harmonic Analysis and Number Theory. Then, an overview follows on the recent results obtained on \mathcal I-characterized subgroups of \mathbb T. In particular, we answer an open question by Ghosh by proving that in case \mathbf u=(u_n)_{n\in\mathbb N} is a sequence of positive integers with u_n\mid u_{n+1} for every n\in\mathbb N and \lim_{n\to\infty}u_n=\infty, and \mathcal I is an ideal of \mathbb N properly containing \mathcal F in, then t_{\mathbf u}(\mathbb T)\subsetneq t_\mathbf u^\mathcal I(\mathbb T).