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 $\left(a_n\right)$) are either countable or of cardinality $c$ depending on the asymptotic behavior of the sequence of the ratios $\frac{a_n}{a_{n-1}}$. 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 ideal Fin. It has been recently established that “Eggleston's dichotomy” fails in the case of modified versions of characterized subgroups when the ideal Fin is replaced by the natural density ideal $I_d$, or more generally, by ideals which are now known as simple density and modular simple density ideals. As all the ideals mentioned above are analytic P-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 snt-ideals). In particular, we are able to establish that for a sequence of positive integers $\left(a_n\right)$, be it arithmetic or arising from the continued fraction expansion of an irrational number:

• (i)

  For non-snt analytic P ideals, the size of the corresponding characterized subgroups is always $c$ even if the sequence $\left(a_n\right)$ is b-bounded (i.e. the sequence of the ratios $\frac{a_n}{a_{n-1}}$ is bounded) which implies the breaking down of “Eggleston's dichotomy”.

• (ii)

  For snt analytic P ideals, the corresponding characterized subgroups are always countable if the sequence $\left(a_n\right)$ is b-bounded which means “Eggleston's dichotomy” holds.