Topologically torsion elements of the circle group for arithmetic-type sequences.

A subgroup of the circle group is called characterized by a sequence of integers if , denoted by whereas the elements of are called topologically torsion elements (corresponding to the sequence ). In this article, we primarily consider a class of sequences which can be extracted from a given arithmetic sequence (we call these extracted sequences “arithmetic-type” sequences) and investigate the corresponding characterized subgroups thoroughly. The primary result of this article is the characterization of topologically torsion elements corresponding to an “arithmetic-type” sequence and thus generalizing the main result of [Dikranjan et al., Comm. Alg. 42 : 600-614, 2014; namely Theorem 2.3] where the characterization of topologically torsion elements for a given arithmetic sequence was established. It is important to note that in the literature there has not been any characterization result for non-arithmetic sequences. This consequently helps us to understand certain cardinality aspects of the characterized subgroups characterized by arithmetic-type sequences all of which happen to be contained in the characterized subgroup characterized by the generating arithmetic sequence.