Search

We make the following three observations regarding a question popularized by Katznelson: is every subset of Z which is a set of Bohr recurrence also a set of topological recurrence? (i) If G is a countable abelian group and E ⊆ G is an I 0 set, then every subset of E - E which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of { 2 n - 2 m : n , m ∈ N } which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let Z ω be the direct sum of countably many copies of Z with standard basis E. If every subset of (E - E) - (E - E) which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime p and let F p ω be the direct sum of countably many copies of Z / p Z with basis (e i) i ∈ N . If for every p-uniform hypergraph with vertex set N and edge set F having infinite chromatic number, the Cayley graph on F p ω determined by { ∑ i ∈ F e i : F ∈ F } has infinite chromatic number, then every subset of F p ω which is a set of Bohr recurrence is a set of topological recurrence. [ABSTRACT FROM AUTHOR]

We provide a necessary and sufficient condition to ensure that a multi-tile Ω ⊂ Rd of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for L2(Ω). New examples are given and this characterization is generalized to abstract locally compact abelian groups. [ABSTRACT FROM AUTHOR]

It is proved that if Ap is a countable elementary abelian p-group, then: (i) The ring End (Ap) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (Ap)/I, where I is the ideal of End (Ap) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (Ap) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained.

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We obtain in this way very sparse sets of real numbers (and of integers) on which two different almost periodic functions cannot agree, which makes them amenable to be used in sampling theorems for these functions. These sets can be made as sparse as to have zero asymptotic density or as to be t-sets, i.e., to be sets that intersect any of their translates in a bounded set. Many of these results are proved not only for almost periodic functions but also for classes of functions generated by more general complex exponential functions, including chirps or polynomial phase functions. [ABSTRACT FROM AUTHOR]

Recall that a space X is selectively pseudocompact if for every sequence { U n : n ∈ N } of non-empty open subsets of X one can choose a point x n ∈ U n for all n ∈ N such that the resulting sequence { x n : n ∈ N } has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by García-Ferreira and Ortiz-Castillo; the present name is due to Dorantes-Aldama and the first listed author. In 2015, García-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact. [ABSTRACT FROM AUTHOR]

We study gaps in the lattice of topological group topologies on a given abelian group and compare the properties of the topologies that form a gap. The emphasis is placed on the study of predecessors of the topology of a noncompact LCA group. It is shown that every Hausdorff predecessor, σ , of the topology τ of an LCA group G is finer than the Bohr topology of the group G and that the two topologies, τ and σ , have the same closed subgroups. This implies that the Bohr topology, τ + , of a noncompact LCA group (G , τ) is not a predecessor of τ. Complementing these results we prove that all predecessors of the topology of a Hausdorff topological abelian group G are Hausdorff provided that G is torsion free. We also show that if a topological abelian group (G , τ) contains a complete subgroup N such that the quotient group G / N is compact, then the predecessors of τ are in a one-to-one correspondence with the predecessors of τ ↾ N on N. In particular, if N is discrete and G / N is compact, then the predecessors of τ are in a one-to-one correspondence with the maximal topological group topologies on N. [ABSTRACT FROM AUTHOR]

This is a survey on the contribution of W. Wistar Comfort to the theory of topological groups through the looking glass of "dense subgroups of topological groups". It turns out that under this umbrella one can find an amazing variety of results, as well as problems posed by Comfort that gave a significant input in the progress in the field of topological groups in the last five decades. [ABSTRACT FROM AUTHOR]

The important rôle that W. W. Comfort played in the study of the Bohr topology is described. [ABSTRACT FROM AUTHOR]

Let X and K be a Čech-complete topological group and a compact group, respectively. We prove that if G is a non-equicontinuous subset of C H o m ( X , K ) , the set of all continuous homomorphisms of X into K , then there is a countably infinite subset L ⊆ G such that L ‾ K X is canonically homeomorphic to βω , the Stone–Čech compactifcation of the natural numbers. As a consequence, if G is an infinite subset of C H o m ( X , K ) such that for every countable subset L ⊆ G and compact separable subset Y ⊆ X it holds that either L ‾ K Y has countable tightness or | L ‾ K Y | ≤ c , then G is equicontinuous. Given a topological group G , denote by G + the (algebraic) group G equipped with the Bohr topology. It is said that G respects a topological property P when G and G + have the same subsets satisfying P . As an application of our main result, we prove that if G is an abelian, locally quasiconvex, locally k ω group, then the following holds: (i) G respects any compact-like property P stronger than or equal to functional boundedness; (ii) G strongly respects compactness. [ABSTRACT FROM AUTHOR]

It is proved that if Ap is a countable elementary abelian p-group, then: (i) The ring End (Ap) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (Ap)/I, where I is the ideal of End (Ap) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (Ap) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained. [ABSTRACT FROM AUTHOR]

Abstract: In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies , we define the equalizer , which permits to describe relevant classes of abelian groups in terms of functorial topologies. [Copyright &y& Elsevier]

Abstract: Let be the set of non-negative integer numbers, the circle group and the cardinality of the continuum. Given an abelian group G of size at most and a countable family of infinite subsets of G, we construct “Baire many” monomorphisms such that is dense in whenever , , and is finite for all and such that for some . We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko (2002) . Applications to group actions and discrete flows on , Diophantine approximation, Bohr topologies and Bohr compactifications are also provided. [Copyright &y& Elsevier]

A topological group is minimally almost periodic (MinAP) if the only continuous homomorphism to any compact group is trivial. Dikranjan and Shakhmatov proved that if an abelian group can be equipped with a MinAP group topology, then for every m ∈ N the subgroup mG of G is either the trivial group or has infinite cardinality. In this paper we prove the following: if an abelian group G can be equipped with a group topology making all of its continuous homomorphic images to a compact group connected, then it admits a MinAP group topology. This condition becomes sufficient as well, as every MinAP topological group only has trivial continuous homomorphic images in compact groups. [ABSTRACT FROM AUTHOR]

A letter to the editor is presented on the invariability of the positive limit set in a continuous discrete system.

Abstract: We prove that every continuous function f with between two bounded Abelian groups G and H equipped with the Bohr topology coincides with a homomorphism when restricted to an infinite subset of the domain. This extends the main results of [K. Kunen, Bohr topology and partition theorems for vector spaces, Topology Appl. 90 (1998) 97–107, D. Dikranjan, S. Watson, A solution to van Douwen''s problem on the Bohr topologies, J. Pure Appl. Algebra 163 (2001) 147–158]. Moreover, we give several applications and we answer a question of [B. Givens, K. Kunen, Chromatic numbers and Bohr topologies, Topology Appl. 131 (2) (2003) 189–202]. [Copyright &y& Elsevier]

Abstract: P is the class of pseudocompact Hausdorff topological groups, and is the class of groups which admit a topology such that . It is known that every is totally bounded, so for the supremum of all pseudocompact group topologies on G and the supremum of all totally bounded group topologies on G satisfy . The authors conjecture for abelian that . That equality is established here for abelian with any of these (overlapping) properties. (a) G is a torsion group; (b) ; (c) ; (d) is a strong limit cardinal, and ; (e) some topology with satisfies ; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and . Furthermore, the product of finitely many abelian , each with the property , has the same property. [Copyright &y& Elsevier]

Abstract: We study a family of idempotent categorical closure operators in the category of topological Abelian groups (and continuous homomorphisms) related to the von Neumann''s kernel. The prominent role is played by the idempotent closure operator also related to questions from Diophantine approximations and ergodic theory. [Copyright &y& Elsevier]

Abstract: For X a compact Abelian group and B an infinite subset of its dual , let be the set of all such that converges to 1. If is a free filter on , let . The sets and are subgroups of X. always has Haar measure 0, while the measure of depends on . We show that there is a filter such that has measure 0 but is not contained in any . This generalizes previous results for the special case where X is the circle group. [Copyright &y& Elsevier]

We use chromatic numbers of hypergraphs to study the Bohr topology G# on discrete Abelian groups. In particular, if K is an infinite Abelian group of a given prime exponent, we show that G# and K# are homeomorphic iff G is the product of K and some finite group. Also, if K is of finite exponent and G is not of finite exponent, then G# is never topologically embeddable in K#. [Copyright &y& Elsevier]

For an infinite abelian group G, let G#denote the Bohr topology on G. We show that G# contains a countable closed subset which is not a retract. This is a generalization of a result due to Gladdines for boolean groups. [Copyright &y& Elsevier]

We consider (discrete) structures, A, for a countable language. A# denotes A with its Bohr topology. Let Y be a compact Hausdorff space. Then Y is homeomorphic to a subspace of some A# iff Y is Talagrand compact. [Copyright &y& Elsevier]

Let P be a property of topological groups. We say that a topological group G is MinAP modulo P if its image f [ G ] under each continuous homomorphism f : G → K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P : finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G / n (G) of every topological group G with respect to its von Neumann kernel n (G) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group (G / n (G)) + equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification. [ABSTRACT FROM AUTHOR]

We show in this paper that for a non-compact LCA group (G , τ) , τ has exactly 2 2 k (G) predecessors in G 2 (G). This answers a problem posed in the literature affirmatively. Denote by N the class of all LCA groups (G , τ) such that P 2 (τ) is precompact. It is shown that for an LCA group G , G ∈ N if and only if G ≅ R n × H , where n is a non-negative integer and H is an LCA group with an open compact subgroup N such that H / N ∈ M. As an application of this result, we extend a well known result on discrete abelian groups to the case of LCA groups. We show that if (G , τ) is a non-compact LCA group and σ is a predecessor of τ in G 2 (G) , then the connected component of (G , τ) coincides with the connected component of (G , σ). It is also shown that for a non-compact LCA group (G , τ) , if σ is a predecessor of τ in G 2 (G) , then the equality i b (G , τ) = i b (G , σ) holds. This partially answer a question posed in the literature. [ABSTRACT FROM AUTHOR]