  Let X be compact abelian group and G its dual (a discrete group). If B is an
infinite subset of G, let C_B be the set of all x in X such that <phi(x) : phi
\in B> converges to 1. If F is a free filter on G, let D_F be the union of all
the C_B for B in 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.
