Countable Discrete Abelian Group Research Articles

Khintchine-type double recurrence in abelian groups

Abstract We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $\Gamma $ is a countable discrete abelian group, $\varphi , \psi \in \mathrm {End}(\Gamma )$ , and $\psi - \varphi $ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $\Gamma $ -system $( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$ , any measurable set $A \in {\mathcal {X}}$ , and any ${\varepsilon }> 0$ , there is a syndetic set of $g \in \Gamma$ such that $\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$ . This generalizes the main results of Ackelsberg et al [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma10 (2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma10 (2022), Paper no. e107]. For the group $\Gamma = {\mathbb {Z}}^d$ , the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations. Forum Math. Sigma10 (2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to the quasi-affine (or Conze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to $\varphi $ and $\psi $ ) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum.

Bohr sets in sumsets II: countable abelian groups

Abstract We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting G be a countable discrete abelian group and $\phi _1, \phi _2, \phi _3: G \to G$ be commuting endomorphisms whose images have finite indices, we show that (1) If $A \subset G$ has positive upper Banach density and $\phi _1 + \phi _2 + \phi _3 = 0$ , then $\phi _1(A) + \phi _2(A) + \phi _3(A)$ contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in $\mathbb {Z}$ and a recent result of the first author. (2) For any partition $G = \bigcup _{i=1}^r A_i$ , there exists an $i \in \{1, \ldots , r\}$ such that $\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$ contains a Bohr set. This generalizes a result of the second and third authors from $\mathbb {Z}$ to countable abelian groups. (3) If $B, C \subset G$ have positive upper Banach density and $G = \bigcup _{i=1}^r A_i$ is a partition, $B + C + A_i$ contains a Bohr set for some $i \in \{1, \ldots , r\}$ . This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices $[G:\phi _j(G)]$ , the upper Banach density of A (in (1)), or the number of sets in the given partition (in (2) and (3)).

Khintchine-type recurrence for 3-point configurations

Abstract The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $\varphi , \psi : G \to G$ are homomorphisms, such that at least two of the three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G, then $\{\varphi , \psi \}$ has the large intersections property. That is, for any ergodic measure preserving system $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ , any $A\in \mathcal {X}$ and any $\varepsilon>0$ , the set $$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$ is syndetic (Theorem 1.11). Moreover, in the special case where $\varphi (g)=ag$ and $\psi (g)=bg$ for $a,b\in \mathbb {Z}$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14). One particularly interesting case is where $G=(\mathbb {Q}_{>0},\cdot )$ and $\varphi (g)=g$ , $\psi (g)=g^2$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $\varphi ,\psi $ that have the large intersections property when $G = {{\mathbb Z}}^2$ . The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages $$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$ In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor. In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $\textbf {X}$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $\sigma $ -algebras of $\varphi (G)$ and $\psi (G)$ invariant functions (Theorem 4.10).

Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $${\mathcal {H}}$$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on $${\mathcal {H}}$$. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group G is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

On Heyde’s Theorem for Probability Distributions on a Discrete Abelian Group

Let X be a countable discrete Abelian group containing no elements of order 2. Let α be an automorphism of X. Let ξ1 and ξ2 be independent random variables with values in the group X and distributions μ1 and μ2. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form L2 = ξ1 + αξ2 given L1 = ξ1 + ξ2 implies that μ j are shifts of the Haar distribution of a finite subgroup of X if and only if α satisfies the condition Ker(I + α)= {0}. Some generalisations of this theorem are also proved.

Association schemes on general measure spaces and zero-dimensional Abelian groups

Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. At the same time, direct extensions of this concept to infinite sets encounter some problems even in the case of countable sets, for instance, countable discrete Abelian groups. In an attempt to resolve these difficulties, we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. However, constructing a complete theory in the general case faces a set of obstacles related to the properties of the adjacency algebras and associated projection operators. To develop a theory of association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. Using the language of spectrally dual partitions, we prove that such groups support the construction of general Abelian (translation) schemes and establish properties of their spectral parameters (eigenvalues).Addressing the existence question of spectrally dual partitions, we show that they arise naturally on topological zero-dimensional Abelian groups, for instance, Cantor-type groups or the groups of p-adic numbers. This enables us to construct large classes of examples of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers (structural constants). We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group.Next we focus on the connection between schemes on zero-dimensional groups and harmonic analysis. We show that the eigenvalues have a natural interpretation in terms of Littlewood–Paley wavelet bases, and in the (equivalent) language of martingale theory. For a class of nonmetric schemes constructed in the paper, the eigenvalues coincide with values of orthogonal function systems on zero-dimensional groups. We observe that these functions, which we call Haar-like bases, have the properties of wavelet bases on the group, including in some special cases the self-similarity property. This establishes a seemingly new link between algebraic combinatorics and (non-Archimedean) harmonic analysis.We conclude the paper by studying some analogs of problems of classical coding theory related to the theory of association schemes.

Topological groups: an introduction

Preface. 1 Groups and Metrics. 1.1 Groups. 1.2 Metric and Topological Spaces. 1.3 Continuous Group Operations. 1.4 Subgroups and Their Quotient Spaces. 1.5 Compactness and Metric Groups. 2 Linear Spaces and Algebras. 2.1 Linear Structures on Groups. 2.2 Linear Functions. 2.3 Norms on Linear Spaces. 2.4 Continuous Linear Functions. 2.5 The Determinant Function. 3 The Subgroups of Rn. 3.1 Closed Subgroups. 3.2 Quotient Groups. 3.3 Dense Subgroups. 4 Matrix Groups. 4.1 General Linear Groups. 4.2 Orthogonal and Unitary Groups. 4.3 Triangular Groups. 4.4 One-Parameter Subgroups. 5 Connectedness of Topological Groups. 5.1 Connected Topological Spaces. 5.2 Connected Matrix Groups. 5.3 Compact Product Spaces. 5.4 Totally Disconnected Groups. 6 Metric Groups of Functions. 6.1 Real-Valued Functions. 6.2 The Compact-Open Topology. 6.3 Metric Groups of Isometries. 6.4 Metric Groups of Homeomorphisms. 6.5 Metric Groups of Homomorphisms. 7 Compact Groups. 7.1 Invariant Means. 7.2 Integral Equations. 7.3 Eigenfunctions. 7.4 Compact Abelian Groups. 7.5 Matrix Representations. 8 Character Groups. 8.1 Countable Discrete Abelian Groups. 8.2 The Duality Homomorphism. 8.3 Compactly Generated Abelian Groups. 8.4 A Duality Theorem. Bibliography. Index of Special Symbols. Index.

HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

AbstractLet X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

The Spectrum of Completely Positive Entropy Actions of Countable Amenable Groups

We prove that an ergodic free action of a countable discrete amenable group with completely positive entropy has a countable Lebesgue spectrum. Our approach is based on the Rudolph–Weiss result on the equality of conditional entropies for actions of countable amenable groups with the same orbits. Relative completely positive entropy actions are also considered. An application to the entropic properties of Gaussian actions of countable discrete abelian groups is given.

The analytic rank of a $C\sp *$-algebra

A concept of rank for unital C*-algebras, which reduces to the dimension of the spectrum in the case of separable abelian C*-algebras, is introduced. This rank has properties somewhat similar to the stable rank, but it appears to be easier to compute in many cases. It is very well behaved with respect to tensor products and crossed products with countable discrete abelian groups

Radon-Nikodym Properties Associated with Subsets of Countable Discrete Abelian Groups

Radon-Nikodym Properties Associated with Subsets of Countable Discrete Abelian Groups

Radon-Nikodým properties associated with subsets of countable discrete abelian groups

With any subset of a countable discrete abelian we associate with it three Banach space properties. These properties are Radon-Nikodym type properties. The relationship between these properties is investigated. The results are applied to give new characterizations of Riesz subsets and Rosenthal subsets of countable discrete abelian groups.