Limits in compact Abelian groups

For X a compact Abelian group and B an infinite subset of its dual X ˆ , let C B be the set of all x ∈ X such that 〈 φ ( x ) : φ ∈ B 〉 converges to 1. If F is a free filter on X ˆ , let D F = ⋃ { C B : B ∈ F } . The sets C B and D F are subgroups of X. C B always has Haar measure 0, while the measure of D F depends on F . We show that there is a filter F such that D F has measure 0 but is not contained in any C B . This generalizes previous results for the special case where X is the circle group.