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1
Content available remote

Self-affine measures that are $L^{p}$-improving

100%
Hare K.
Colloquium Mathematicum
|
2015
|
tom 139
|
nr 2
229-243
EN
A measure is called $L^{p}$-improving if it acts by convolution as a bounded operator from $L^{q}$ 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 $L^{p}$-improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$-improving.
2
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Distinctness of spaces of Lorentz-Zygmund multipliers

64%
Hare K., Mohanty P.
Studia Mathematica
|
2005
|
tom 169
|
nr 2
143-161
EN
We study the spaces of Lorentz-Zygmund multipliers on compact abelian groups and show that many of these spaces are distinct. This generalizes earlier work on the non-equality of spaces of Lorentz multipliers.
3
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Exact Kronecker constants of Hadamard sets

64%
Hare K., Ramsey L.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 1
39-49
EN
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, $α{1,m,...,m^{d-1}} = (m^{d-1}-1)/(2(m^{d}-1))$ and α{1,m,m²,...} = 1/(2m).
4
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ε-Kronecker and I₀ sets in abelian groups, III: interpolation by measures on small sets

64%
Graham C., Hare K.
Studia Mathematica
|
2005
|
tom 171
|
nr 1
15-32
EN
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 and ε-Kronecker sets, and a slightly weaker general form when Γ has torsion. This extends previously known results for Sidon, ε-Kronecker, and Hadamard sets.
5
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A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

64%
Hare K., Yamagishi S.
Acta Arithmetica
|
2014
|
tom 166
|
nr 1
55-67
EN
Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_{N}^{(m)}(ω)$ 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 $r_{N}^{(m)}(ω) < K$ 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.
6
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Characterizing Sidon sets by interpolation properties of subsets

64%
Graham C., Hare K.
Colloquium Mathematicum
|
2008
|
tom 112
|
nr 2
175-199
EN
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 but a finite number of elements of order 2.
7
Content available remote

On convolution squares of singular measures

64%
Gupta S., Hare K.
Colloquium Mathematicum
|
2004
|
tom 100
|
nr 1
9-16
EN
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.
8
Content available remote

ε-Kronecker and I₀ sets in abelian groups, IV: interpolation by non-negative measures

64%
Graham C., Hare K.
Studia Mathematica
|
2006
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tom 177
|
nr 1
9-24
EN
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).
9
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$L^{p}$-improving properties of measures of positive energy dimension

64%
Hare K., Roginskaya M.
Colloquium Mathematicum
|
2005
|
tom 102
|
nr 1
73-86
EN
A measure is called $L^{p}$-improving if it acts by convolution as a bounded operator from $L^{p}$ to $L^{q}$ for some q > p. Positive measures which are $L^{p}$-improving are known to have positive Hausdorff dimension. We extend this result to complex $L^{p}$-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 $L^{p}$-functions.
10
Content available remote

Energy of measures on compact Riemannian manifolds

64%
Hare K., Roginskaya M.
Studia Mathematica
|
2003
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tom 159
|
nr 2
291-314
EN
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 dimension of the measure. Refined energy integrals and Hausdorff dimensions are also studied and applied to investigate the singularity of Riesz product measures of dimension one.
11
Content available remote

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

64%
Graham C., Hare K.
Colloquium Mathematicum
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2012
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tom 127
|
nr 1
1-15
EN
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 → 𝕋 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" elements of order 2 in the subgroup generated by E. When there are "too many" elements of order 2, we show that there exists a subset F, of the same cardinality as E, on which every {-1,1}-valued function can be interpolated exactly. Such sets are also I₀. In both cases, the set F also has the property that the only continuous character at which $F·F^{-1}$ can cluster in the Bohr topology is 1. This improves upon previous results concerning the existence of I₀ subsets of a given E.
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