On the Borel complexity of characterized subgroups

In a compact abelian group X, a subgroup H   is called characterized if there exists a sequence of characters v=(vn)v=(vn) of X   such that H={x∈X:vn(x)→0 in T}H={x∈X:vn(x)→0 in T}. Gabriyelyan proved for X=TX=T, that the characterized subgroup {x∈T:n!x→0 in T}{x∈T:n!x→0 in T} is not an FσFσ-set. In this paper, we obtain a complete description of the FσFσ-subgroups of TT characterized by sequences of integers v=(vn)v=(vn) such that vn|vn+1vn|vn+1 for all n∈Nn∈N showing that these are exactly the countable subgroups of TT. Moreover, in the general setting of compact metrizable abelian groups, we give a new point of view to study the Borel complexity of characterized subgroups in terms of appropriate test-topologies defined on the group X.