Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and f : H → T = R / Z a (not necessarily continuous) character of H. Then there exists a sequence ( χ n ) n = 1 ∞ of (continuous) characters of G such that lim n → ∞ χ n ( α ) = f ( α ) for all α ∈ H and ( χ n ( α ) ) n = 1 ∞ does not converge whenever α ∈ G ∖ H . If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.