On topologically torsion elements and their subgroups corresponding to arithmetic-type sequences.

An element of the circle group is said to be topologically torsion with respect to a sequence of integers if . The most well-established results concerning such elements are primarily associated with arithmetic sequences.

In this study, we extend the investigation to arithmetic-type sequences, conducting a comprehensive analysis of these elements and their corresponding characterized subgroups, with particular attention to their cardinality aspects. Since arithmetic sequences constitute a special subclass of arithmetic-type sequences, our findings provide a more generalized framework, unveiling broader structural properties that encompass and refine several intriguing results known for arithmetic sequences.