Abstract
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)nâN0, with un|un+1 for all nâN0, such that H consists precisely of those elements xâT with unxâ0 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.
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