A brief history and recent advances in the theory of characterized subgroups of the circle group

Abstract: A subgroup $H$ of the circle group $\mathbb T$ is said to be characterized by a sequence of integers $\mathbf u = (u_n)_{n\in\mathbb N}$ if $H=\{x\in\mathbb T: u_nx\to 0\}$. The first part of the talk discusses 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 (topologically torsion elements and characterized subgroups), Harmonic Analysis (sets of convergence of trigonometric series, thin sets) and Number Theory (uniform distribution of sequences).

Recently, generalizations of the notion of characterized subgroup of $\mathbb T$ were introduced, based on weaker notions of convergence, starting from statistical convergence and ending with $\mathcal I$-convergence for an ideal $\mathcal I$ of $\mathbb N$, due to Cartan. 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 $\mathbb T$. A subgroup $H$ of the circle group $\mathbb T$ is said to be $\mathcal I$-characterized with respect to $\mathcal I$ by a sequence of integers $\mathbf u = (u_n)_{n\in\mathbb N}$ if $H=\{x\in\mathbb T: u_nx\overset{\mathcal I}\to 0\}$. The second part of the presentation proposes an overview on the results obtained on these new kind of characterized subgroups, with special emphasis on $\mathcal I$-characterized subgroups of $\mathbb T$.

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

group theory

Audience: researchers in the topic

Groups in Galway 2024