A characterization of the maximally almost periodic abelian groups

Abstract

We introduce a categorical closure operator g in the category of topological abelian groups (and continuous homomorphisms) as a Galois closure with respect to an appropriate Galois correspondence defined by means of the Pontryagin dual of the underlying group. We prove that a topological abelian group G is maximally almost periodic if and only if every cyclic subgroup of G is g-closed. This generalizes a property characterizing the circle group from (Studia Sci. Math. Hungar. 38 (2001) 97-113. A characterization of the circle group and the p-adic integers via sequential limit laws, preprint). and answers an appropriate version of a question posed in (A characterization of the circle group and the p-adic integers via sequential limit laws, preprint)

Dirichlet sets vs characterized subgroups

“…It will be immediately clear from the definitions that a subset A of T is an A-set precisely when A is contained in some characterized subgroup of T. Indeed, following [20] (the terminology was given in [7]), a subgroup H of T is characterized if there exists a sequence u in Z such that H coincides with t u (T) := {x ∈ T : u n x → 0}.…”

Introduction to topological groups

“…The above theorem suggested to use in [35] a different approach to the problem, replacing the sequence of integers u n (characters of T!) by a sequence u n in the Pontryagin-van Kampen dual G. Then the subgroup s u (G) = {x ∈ G : lim n u n (x) = 0 in T} of G really can be used for such a characterization of all countable subgroups of the compact metrizable groups (see [35,33,17] for major detail).…”

Topological torsion related to some recursive sequences of integers

“…On the other hand, it was proved by Borel [11] that every countable subgroup H of T has the form t u (T) for an approriate sequence u (see also [9]). This result was extended in appropriate way to compact metrizable abelian groups in [19,7] (see also [8,10,20,18] for related results). This paper is dedicated to the study of the subgroups t u (T) of T for sequences u := (u n ) n≥1 of integers that verify a linear recurrence relation of order ≤ k, u n = a (1) n u n−1 + a (2) n u n−2 + .…”