Raffaele di Santo

Element-wise description of the \mathcal{I}-characterized subgroups of the circle.

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 {\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 \ \mathcal {I}-converges to 0 \}.

This set is a 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). 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 from [4].

References

[1] P. Das, A. Ghosh, Solution of a general version of Armacost’s problem on topologically torsion elements, Acta Math. Hungar. 164 (2021), no. 1, 243—264.

[2] R. Di Santo, D. Dikranjan, A. Giordano Bruno, H. Weber, Element-wise description of the {\mathcal{I}}-characterized subgroups of the circle, submitted.

[3] R. Di Santo, D. Dikranjan, A. Giordano Bruno, H. Weber, Nested ideals and topologically {\mathcal{I}}-torsion elements of the circle group, submitted.

[4] D. Dikranjan, D. Impieri, Topologically torsion elements of the circle group, Comm. Algebra 42 (2014), 600—614.

[5] A. Ghosh, Topologically {\mathcal{I}}-torsion elements of the circle, Ric. Mat. 73 (2024) no. 4, 2263—2281.