Wyniki wyszukiwania

1

ε-Kronecker and I₀ sets in abelian groups, III: interpolation by measures on small sets

100%

Graham C., Hare K.

Studia Mathematica

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2005

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tom 171

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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.

2

Characterizing Sidon sets by interpolation properties of subsets

100%

Graham C., Hare K.

Colloquium Mathematicum

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2008

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tom 112

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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.

3

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

100%

Graham C., Hare K.

Studia Mathematica

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2006

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tom 177

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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).

4

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

100%

Graham C., Hare K.

Colloquium Mathematicum

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2012

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tom 127

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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.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Studia Mathematica

4 Graham C.

4 Hare K.

1 2012

1 2008

1 2006

1 2005

ε-Kronecker and I₀ sets in abelian groups, III: interpolation by measures on small sets

Characterizing Sidon sets by interpolation properties of subsets

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

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