Dirichlet sets vs characterized subgroups

Abstract

A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u=(un)n∈N0 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. Finally, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.