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

According to Cartan, given an ideal (Formula presented.) of (Formula presented.), a sequence (Formula presented.) in the circle group (Formula presented.) is said to (Formula presented.) -converge to a point (Formula presented.) if (Formula presented.) for every neighborhood U of x in (Formula presented.). For a sequence (Formula presented.) in (Formula presented.), let (Formula presented.) When (Formula presented.) is an analytic free P-ideal, this set (Formula presented.) is a Borel (hence, Polishable) subgroup of (Formula presented.) with many nice properties. It has been largely studied in the case when (Formula presented.) is the ideal (Formula presented.) of all finite subsets of (Formula presented.) (so (Formula presented.) -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 (Formula presented.) when (Formula presented.) for every (Formula presented.) and under suitable hypotheses on (Formula presented.). In the special case when (Formula presented.), we obtain an alternative proof of a simplified equivalent version of a known result from [18].