Generating subgroups of the circle using statistical convergence of order α

Generating Subgroups of the Circle using a Generalized class of Density Functions

“…As mentioned already, all the notions of density functions, precisely, natural density d [13], natural density of order α d α [6], their generalization with respect to weight function g, d g [5] and natural density with respect to an unbounded modulus function f , fdensity d f [1] are special cases of "f density of weight g" i.e. d f g [8] and the consequence is that, not only the main results of [19] and [9] follow as special cases of our results, namely, Theorem 3.4 (extends Theorem A [19]), Theorem 3.12 (extending Theorem B [19]) and Theorem 3.13 (extending Theorem C [19]), at the same time, the questions about f -density are resolved. Finally the justification for the investigation is assured by Theorem 4.1 and Theorem 4.2 (proved in Section 4) which shows that for a given arithmetic sequence, we can indeed construct nontrivial Borel subgroups of T different from t s (an) (T) or t α (an) (T) for suitable choice of modulus function f or the weight function g. The article ends with some interesting comparative results about the generated subgroups which are also presented in this section.…”

“…In the recent article [9] the following open problem was posed: Problem 2.1. For any arithmetic sequence (a n ) and 0 < α 1 < α 2 < 1, is t α1 (an) (T) t α2 (an) (T) ?…”

“…This was followed by another attempt to generate nice subgroups of T using certain kind of density function when in [9] the notion of natural density of order α, 0 < α < 1, (introduced in 2012 in [6]) was used to generate corresponding characterized subgroups t α (an) (T) and it was seen that they indeed generate, for a given arithmetic sequence (a n ), new nontrivial subgroups different from t (an) (T) as well as t s (an) (T). Lately there have been certain other as also more general versions of density functions which we now recall.…”