Sets of zero discrete harmonic density

2009 ◽  
Vol 148 (2) ◽  
pp. 253-266 ◽  
Author(s):  
COLIN C. GRAHAM ◽  
KATHRYN E. HARE
Keyword(s):  
Abelian Group ◽  
Dual Group ◽  
New Approach ◽  
Sidon Sets ◽  
Harmonic Density

AbstractLet G be a compact, connected, abelian group with dual group Γ. The set E ⊂ has zero discrete harmonic density (z.d.h.d.) if for every open U ⊂ G and μ ∈ Md(G) there exists ν ∈ Md(U) with = on E. I0 sets in the duals of these groups have z.d.h.d. We give properties of such sets, exhibit non-Sidon sets having z.d.h.d., and prove union theorems. In particular, we prove that unions of I0 sets have z.d.h.d. and provide a new approach to two long-standing problems involving Sidon sets.

scholarly journals The Relationship Between ϵ-Kronecker Sets and Sidon Sets

2016 ◽  
Vol 59 (3) ◽  
pp. 521-527 ◽  
Author(s):  
Kathryn Hare ◽  
L. Thomas Ramsey
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Dual Group ◽  
Sidon Sets ◽  
Sidon Set ◽  
Arithmetic Properties ◽  
The Fourier Transform ◽  
The Relationship ◽  
Discrete Abelian Group

AbstractA subset E of a discrete abelian group is called ϵ-Kronecker if all E-functions of modulus one can be approximated to within ϵ by characters. E is called a Sidon set if all bounded E-functions can be interpolated by the Fourier transform of measures on the dual group. As ϵ-Kronecker sets with ϵ < 2 possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

scholarly journals Coordinates for analytic operator algebras

1989 ◽  
Vol 31 (1) ◽  
pp. 31-47
Author(s):  
Baruch Solel
Keyword(s):  
Abelian Group ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Dual Group ◽  
Positive Semigroup ◽  
Analytic Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Analytic Algebra ◽  
Image Position

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].

scholarly journals Rigidity theory for C∗-dynamical systems and the “Pedersen rigidity problem”

2018 ◽  
Vol 29 (03) ◽  
pp. 1850016 ◽  
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Dynamical Systems ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Alternative Formulation ◽  
Crossed Products ◽  
Locally Compact ◽  
Fixed Point Algebra ◽  
Multiplier Algebras

Let [Formula: see text] be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of [Formula: see text] on [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of [Formula: see text] and [Formula: see text] in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of [Formula: see text] and [Formula: see text]. There is an alternative formulation of the problem: an action of the dual group [Formula: see text] together with a suitably equivariant unitary homomorphism of [Formula: see text] give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of [Formula: see text] and two such equivariant unitary homomorphisms of [Formula: see text] that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of [Formula: see text] and [Formula: see text] is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if [Formula: see text] is discrete, this will be the case for all actions of [Formula: see text].

scholarly journals Multipliers for Amalgams and the Algebra S0(G)

1987 ◽  
Vol 39 (1) ◽  
pp. 123-148 ◽  
Author(s):  
Maria L. Torres De Squire
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
The Difference ◽  
Image Position

Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,For a function f on G and s ∊ G, the functions f′ and fs will be defined by

On Multipliers with Unconditionally Converging Fourier Series

1972 ◽  
Vol 24 (3) ◽  
pp. 477-484 ◽  
Author(s):  
Gregory F. Bachelis ◽  
Louis Pigno
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Complex Valued

Let G be a compact abelian group with dual group Γ. For 1 ≦ p < ∞, 1 ≦ q < ∞, let denote the Banach space of complex-valued functions on Γ which are multipliers of type (p, q) and the subspace of compact multipliers.Grothendieck [10; 11] has proven that a function in LP(G), 1 ≦ p < 2, has an unconditionally converging Fourier series in LP(G) if and only if it is in L2(G), and Helgason [12] has proven that the derived algebra of LP(G), 1 ≦ p < 2, is L2(G). Using these results we show in § 2 that a multiplier of type (p, g), 1 ≦ p ≦ 2, 1 ≦ q ≦ 2, has an unconditionally converging Fourier series in if and only if it is in (Theorem 2.1), and that, for 1 ≦ p ≦ q ≦ 2, the derived algebra of is (Theorem 2.2). Statements equivalent to the above are also given.

Gap Series on Groups and Spheres

1966 ◽  
Vol 18 ◽  
pp. 389-398 ◽  
Author(s):  
Daniel Rider
Keyword(s):  
Abelian Group ◽  
Trigonometric Polynomial ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Gap Series ◽  
Image Position

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E wheredx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.

scholarly journals Union results for thin sets

1990 ◽  
Vol 32 (2) ◽  
pp. 241-254
Author(s):  
Kathryn E. Hare
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Group Operation ◽  
Borel Measures ◽  
Thin Sets

Let G be a compact abelian group and let Γ be its (discrete) dual group. Denote by M(G) the space of complex regular Borel measures on G.Let E be a subset of Γ. Then:(i) E is called a Rajchman set if, for all μ ∈M(G) implies (ii) E is called a set of continuity if given ε > 0 there exists δ > 0 such that if and(iii) E is called a parallelepiped of dimension N if |E| = 2N and there are two-element sets . (The multiplication indicated here is the group operation.)

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