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

Colloquium Mathematicae (2012)

• Volume: 127, Issue: 1, page 1-15
• ISSN: 0010-1354

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## Abstract

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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” 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.

## How to cite

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Colin C. Graham, and Kathryn E. Hare. "Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups." Colloquium Mathematicae 127.1 (2012): 1-15. <http://eudml.org/doc/283584>.

@article{ColinC2012,
abstract = {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” 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.},
author = {Colin C. Graham, Kathryn E. Hare},
journal = {Colloquium Mathematicae},
keywords = {-Kronecker sets; Fatou-Zygmund property; -free sets; Hadamard sets; sets; Sidon sets},
language = {eng},
number = {1},
pages = {1-15},
title = {Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups},
url = {http://eudml.org/doc/283584},
volume = {127},
year = {2012},
}

TY - JOUR
AU - Colin C. Graham
AU - Kathryn E. Hare
TI - Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups
JO - Colloquium Mathematicae
PY - 2012
VL - 127
IS - 1
SP - 1
EP - 15
AB - 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” 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.
LA - eng
KW - -Kronecker sets; Fatou-Zygmund property; -free sets; Hadamard sets; sets; Sidon sets
UR - http://eudml.org/doc/283584
ER -

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