Factorizable subgroups of the circle group

A subgroup H of the circle group T is said to be a-characterized if there exists a strictly increasing sequence of positive integers (un)n∈N0, with un|un+1 for all n∈N0, such that H consists precisely of those elements x∈T with unx→0 in T. These subgroups appeared in the study of trigonometric series in harmonic analysis, as well as in Diophantine approximation, dynamical systems and ergodic theory. The aim of the paper is to show that any a-characterized subgroup of T can be presented as the sum of two of its proper a-characterized subgroups.