Element-wise description of the I -characterized subgroups of the circle

According to Cartan, given an ideal I of N, a sequence (xn)n∈N in the circle group T is said to I-converge to a point x∈T if {n∈N:xn∉U}∈I for every neighborhood U of x in T. For a sequence u=(un)n∈N in Z, let tuI(T):={x∈T:unx I‐converges to 0}. When I is an analytic free P-ideal, this set tuI(T) is a Borel (hence, Polishable) subgroup of T with many nice properties. It has been largely studied in the case when I is the ideal Fin of all finite subsets of N (so Fin-convergence coincides with the usual notion of convergence), due to its remarkable connections to topological algebra, descriptive set theory and harmonic analysis. In this paper, we give a complete element-wise description of tuI(T) when un|un+1 for every n∈N and under suitable hypotheses on I. In the special case when I=Fin, we obtain an alternative proof of a simplified equivalent version of a known result from [Citation18].