Characterizing subgroups of compact abelian groups

Dikranjan, Dikran; Kunen, Kenneth

doi:10.1016/j.jpaa.2005.12.011

Cited by 29 publications

(35 citation statements)

References 24 publications

(45 reference statements)

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“…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 + .…”

Section: Introductionmentioning

confidence: 94%

Topological torsion related to some recursive sequences of integers

Barbieri

Dikranjan

Milan

et al. 2008

Mathematische Nachrichten

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For a recursively defined sequence u := (un) of integers, we describe the subgroup tu(T) of the elements x of the circle group T satisfying limn unx = 0. More attention is dedicated to the sequences satisfying a secondorder recurrence relation. In this case, we show that the size and the free-rank of tu(T) is determined by the asymptotic behaviour of the ratios qn = un u n−1 and we extend previous results of G. Larcher, C. Kraaikamp, and P. Liardet obtained from continued fraction expansion.

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Section: Introductionmentioning

confidence: 94%

Topological torsion related to some recursive sequences of integers

Barbieri

Dikranjan

Milan

et al. 2008

Mathematische Nachrichten

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“…If u is not eventually 0, we suppose without loss of generality that all members of u are non-zero, so we can also assume that u is in N, since t u (T) = t |u| (T), where |u| = (|u n |) n∈N 0 . It is worth to recall also that every countable subgroup of T is characterized (see [7], and see [6] and [19] for more general results).…”

Section: A-set)mentioning

confidence: 99%

Dirichlet sets vs characterized subgroups

Barbieri

Dikranjan

Bruno

et al. 2017

Topology and its Applications

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MSC: Keywords:Characterized subgroup a-sequence of integers Circle group Dirichlet set Arbault set Factorizable subgroup A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u = (u n ) n∈N0 in N such that u n x → 0 uniformly on A. In particular, A is contained in the subgroup t u (T) := {x ∈ T : u n x → 0}, which is the subgroup of T characterized by u. Using strictly increasing sequences u in N such that u n divides u n+1 for every n ∈ N, we find in T a family of closed perfect D-sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D-sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Finally, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.

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“…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).…”

Section: Precompact Group Topologies Determined By Sequencesmentioning

confidence: 99%