Inclusions of characterized subgroups

A subgroup H of R is characterized if H=τu(R):={x∈R:unx→0modZ} for some sequence u in R. Given two sequences u and v in R, we find conditions under which τu(R) is contained or not in τv(R). As a by-product of our main theorems, we find a known result by Eliaš on inclusions of characterized subgroups of T, motivated by problems in harmonic analysis.