Limits in compact abelian groups
Abstract
Let X be compact abelian group and G its dual (a discrete group). If B is an infinite subset of G, let CB be the set of all x in X such that <phi(x) : phi ∈ B> converges to 1. If F is a free filter on G, let DF be the union of all the CB for B in F. The sets CB and DF are subgroups of X. CB always has Haar measure 0, while the measure of DF depends on F. We show that there is a filter F such that DF has measure 0 but is not contained in any CB. This generalizes previous results for the special case where X is the circle group.
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