Eggleston’s dichotomy for characterized subgroups and the role of ideals

“Eggleston’s dichotomy” is a “one of a kind” unique observation which broadly tells us that the characterized subgroups of the circle group (characterized by a sequence of positive integers ) are either countable or of cardinality depending on the asymptotic behavior of the sequence of the ratios . One should note that these subgroups are generated by using the notion of usual convergence which is nothing but a special case of the more general notion of ideal convergence for the Frechet ideal . It has been recently established that “Eggleston’s dichotomy” fails in the case of modified versions of characterized subgroups when the ideal is replaced by the natural density ideal or more generally ideals which are now known as simple density and modular simple density ideals. As the ideals mentioned above are analytic -ideals, a natural question arises as to whether one can isolate some appropriate property of ideals which enforces the dichotomy or the failure of it. In this article we are able to isolate that particular feature of an ideal and come out with a new class of ideals which we call, “strongly non-translation invariant ideals” (in short -ideals). In particular, we are able to establish that for an arithmetic sequence of positive integers ,
(i) For non- analytic ideals, the size of the corresponding characterized subgroups is always even if the sequence is -bounded (i.e. the sequence of the ratios is bounded) and so “Eggleston’s dichotomy” fails.
(ii) For analytic ideals, the corresponding characterized subgroups are countable if the sequence is -bounded which means “Eggleston’s dichotomy” holds.