Sidonicity in compact, abelian hypergroups
Random weighted Sidon sets
We investigate random Sidon-type sets in which the degrees of the representations are weighted. These variants of Sidon sets are of interest as there are compact non-abelian groups which admit no infinite Sidon sets. In this note we determine the largest weight function such that infinite random weighted Sidon sets exist in all infinite compact groups.
Tame -multipliers
We call an -multiplier m tame if for each complex homomorphism χ acting on the space of multipliers there is some and |a| ≤ 1 such that for all γ ∈ Γ. Examples of tame multipliers include tame measures and one-sided Riesz products. Tame multipliers show an interesting similarity to measures. Indeed we show that the only tame idempotent multipliers are measures. We obtain quantitative estimates on the size of -improving tame multipliers which are similar to those obtained for measures, but...
The size of multipliers
The support of a function with thin spectrum
We prove that if does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty there exists a constant c > 0 such that for all whose Fourier transform is supported on E. In particular, such functions cannot vanish on any open, non-empty subset of G. Examples of sets which do not contain parallelepipeds of arbitrarily large dimension include all Λ(p) sets.
The size of characters of compact Lie groups
Pointwise upper bounds for characters of compact, connected, simple Lie groups are obtained which enable one to prove that if μ is any central, continuous measure and n exceeds half the dimension of the Lie group, then . When μ is a continuous, orbital measure then is seen to belong to . Lower bounds on the p-norms of characters are also obtained, and are used to show that, as in the abelian case, m-fold products of Sidon sets are not p-Sidon if p < 2m/(m+1).
Completely bounded lacunary sets for compact non-abelian groups
In this paper, we introduce and study the notion of completely bounded sets ( for short) for compact, non-abelian groups G. We characterize sets in terms of completely bounded multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are sets for all p < ∞, but are not for any p ≥ 4. This is done by showing that the space of completely bounded multipliers...
Strict-2-associatedness for thin sets
Self-affine measures that are -improving
A measure is called -improving if it acts by convolution as a bounded operator from to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be -improving.
Central sidonicity for compact Lie groups
It is known that the dual of a compact, connected, non-abelian group may contain no infinite central Sidon sets, but always does contain infinite central -Sidon sets for We prove, by an essentially constructive method, that the latter assertion is also true for every infinite subset of the dual. In addition, we investigate the relationship between weighted central Sidonicity for a compact Lie group and Sidonicity for its torus.
Applicationsof Generalized Perron Trees to Maximal Functions and Density Bases.
A generalization of a theorem of Erdős-Rényi to m-fold sums and differences
Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.
-improving properties of measures of positive energy dimension
A measure is called -improving if it acts by convolution as a bounded operator from to for some q > p. Positive measures which are -improving are known to have positive Hausdorff dimension. We extend this result to complex -improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of -functions.
On convolution squares of singular measures
We prove that for every compact, connected group G there is a singular measure μ such that the Fourier series of μ*μ converges uniformly on G. Our results extend the earlier results of Saeki and Dooley-Gupta.
Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups
Let G be a compact abelian group with dual group Γ and let ε > 0. A set E ⊂ Γ is a “weak ε-Kronecker set” if for every φ:E → there exists x in the dual of Γ such that |φ(γ)- γ(x)| ≤ ε for all γ ∈ E. When ε < √2, every bounded function on E is known to be the restriction of a Fourier-Stieltjes transform of a discrete measure. (Such sets are called I₀.) We show that for every infinite set E there exists a weak 1-Kronecker subset F, of the same cardinality as E, provided there are not “too many”...
Exact Kronecker constants of Hadamard sets
A set S of integers is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant, κ(S). The angular Kronecker constant is the unique real number α(S) ∈ [0,1/2] such that κ(S) = |exp(2πiα(S)) - 1|. We show that for integers m > 1 and d ≥ 1, and α1,m,m²,... = 1/(2m).
Characterizing Sidon sets by interpolation properties of subsets
Pisier's characterization of Sidon sets as containing proportional-sized quasi-independent subsets is given a sharper form for groups with only a finite number of elements having orders a power of 2. No such improvement is possible for a general Sidon subset of a group having an infinite number of elements of order 2. The method used also gives several sharper forms of Ramsey's characterization of Sidon sets as containing proportional-sized I₀-subsets in a uniform way, again in groups containing...
ε-Kronecker and I₀ sets in abelian groups, III: interpolation by measures on small sets
Let U be an open subset of a locally compact abelian group G and let E be a subset of its dual group Γ. We say E is I₀(U) if every bounded sequence indexed by E can be interpolated by the Fourier transform of a discrete measure supported on U. We show that if E·Δ is I₀ for all finite subsets Δ of a torsion-free Γ, then for each open U ⊂ G there exists a finite set F ⊂ E such that E∖F is I₀(U). When G is connected, F can be taken to be empty. We obtain a much stronger form of that for Hadamard sets...
Energy of measures on compact Riemannian manifolds
We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted L² norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff...
ε-Kronecker and I₀ sets in abelian groups, IV: interpolation by non-negative measures
A subset E of a discrete abelian group is a "Fatou-Zygmund interpolation set" (FZI₀ set) if every bounded Hermitian function on E is the restriction of the Fourier-Stieltjes transform of a discrete, non-negative measure. We show that every infinite subset of a discrete abelian group contains an FZI₀ set of the same cardinality (if the group is torsion free, a stronger interpolation property holds) and that ε-Kronecker sets are FZI₀ (with that stronger interpolation property). ...
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