Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups

Colin C. Graham; Kathryn E. Hare

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Displaying similar documents to “Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups”

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan (2013)

Fundamenta Mathematicae

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Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G, τ)}^{\land}$ is a dense -closed subgroup of the compact group ${(G_{d})}^{\land}$ , where $G_{d}$ is the group G with...

The dual group of a dense subgroup

William Wistar Comfort, S. U. Raczkowski, F. Javier Trigos-Arrieta (2004)

Czechoslovak Mathematical Journal

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Throughout this abstract, $G$ is a topological Abelian group and $\hat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\hat{G} ↠ \hat{D}$ given by $h \mapsto h | D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$ . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable...

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

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Let G be an additive finite abelian group. For every positive integer ℓ, let $d i s c_{ℓ} (G)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $d i s c_{ℓ} (G)$ for certain finite groups, including cyclic groups, the groups $G = C ₂ \oplus C_{2 m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum...

Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort, Lewis C. Robertson

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For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly...

Two examples of subspaces in $L^{2 l}$ spanned by characters of finite order

Mats Erik Andersson (2001)

Colloquium Mathematicae

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By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into $L^{2 l}$ of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.

On the vector-valued Fourier transform and compatibility of operators

In Sook Park (2005)

Studia Mathematica

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Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform $F^{}$ if $F^{} \otimes T : L_{p} () \otimes X \to L_{p^{'}} (^{'}) \otimes Y$ admits a continuous extension $[F^{}, T] : [L_{p} (), X] \to [L_{p^{'}} (^{'}), Y]$ . Let $ℱ T_{p}^{}$ denote the collection of such T’s. We show that $ℱ T_{p}^{ℝ \times} = ℱ T_{p}^{ℤ \times} = ℱ T_{p}^{ℤ ⁿ \times}$ for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then...

A note on a class of factorized $p$ -groups

Enrico Jabara (2005)

Czechoslovak Mathematical Journal

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In this note we study finite $p$ -groups $G = A B$ admitting a factorization by an Abelian subgroup $A$ and a subgroup $B$ . As a consequence of our results we prove that if $B$ contains an Abelian subgroup of index $p^{n - 1}$ then $G$ has derived length at most $2 n$ .

Sidon sets and $I^{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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$ℵ_{k}$ -free separable groups with prescribed endomorphism ring

Daniel Herden, Héctor Gabriel Salazar Pedroza (2015)

Fundamenta Mathematicae

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We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of $ℵ_{k}$ -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is $ℵ_{k}$ -free if every subset of size $< ℵ_{k}$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a...

An $L^{p}$ -version of a theorem of D.A. Raikov

Gero Fendler (1985)

Annales de l'institut Fourier

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Let $G$ be a locally compact group, for $p \in (1, \infty)$ let $P f_{p} (G)$ denote the closure of $L^{1} (G)$ in the convolution operators on $L^{p} (G)$ . Denote $W_{p} (G)$ the dual of $P f_{p} (G)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_{p} (G)$ . It is shown that on the unit sphere of $W_{p} (G)$ the $σ (W_{p}, P f_{p})$ topology and the strong $A_{p}$ -multiplier topology coincide.

On functions whose translates are independent

Ralph E. Edwards (1951)

Annales de l'institut Fourier

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Ce travail est l’étude de divers cas particuliers d’un problème nouveau, semble-t-il, concernant les translatées de fonctions ou de distributions sur un groupe. Soit $E$ un espace vectoriel topologique de fonctions ou de distributions sur un groupe abélien $G$ localement compact ; $E$ est supposé invariant par les translations $a \to f_{a} (x) = f (x + a) (f \in E, a \in G)$ . Si $f \in E$ et si $A$ est un sous-ensemble non vide de $G$ , $I (f, A) = I (f, A, E)$ désigne le sous-espace vectoriel fermé de $E$ engendré par les translatées $f_{a}$ de $f$ avec $a \in A$ . On dira qu’une $f \in E$ a ses...