  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 (chi_n)_n of (continuous) characters of G such that lim_n chi_n(alpha)
= f(alpha) for all alpha in H and (chi_n(alpha))_n does not converge whenever
alpha in 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.
