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 (u(n)) n is an element of N-0, with u(n) |u(n+1) for all n is an element of N-0, such that H consists precisely of those elements x is an element of T with u(n)x -> 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. (c) 2022 Elsevier B.V. All rights reserved.