Generating subgroups of the circle using a generalized class of density functions

Abstract

In this article, we consider the generalized version dgf of the natural density function introduced in Bose et al. (2018) where g:N→[0,∞) satisfies g(n)→∞ and ng(n)↛0 whereas f is an unbounded modulus function and generate versions of characterized subgroups of the circle group T using these density functions. We show that these subgroups have the same feature as the s-characterized subgroups (Dikranjan et al., 2020) or α-characterized subgroups (Bose et al., 2020) and our results provide more general versions of the main results of both the articles. But at the same time the utility of this more general approach is justified by constructing new and nontrivial subgroups for suitable choice of f and g. In several of our results we use properties of the ideal Zg(f) which are first presented along with certain new observations about these ideals which were not there in Bose et al. (2018).