Dirichlet sets vs Characterized subgroups

A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u = (un) n∈N 0 in N such that unx → 0 uniformly on A. In particular, A is contained in the subgroup tu(T) := {x ∈ T : unx → 0}, which is the subgroup of T characterized by u. Using strictly increasing sequences u in N such that un divides un+1 for every n ∈ N, we find in T a family of closed perfect D-sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D-sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Moreover, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.