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 can cluster in the Bohr topology is 1. This improves upon previous results concerning the existence of I₀ subsets of a given E.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm127-1-1,
author = {Colin C. Graham and Kathryn E. Hare},
title = {Existence of large e-Kronecker and FZI0(U) sets in discrete abelian groups},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {1-15},
zbl = {1277.43010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm127-1-1}
}
Colin C. Graham; Kathryn E. Hare. Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups. Colloquium Mathematicae, Tome 126 (2012) pp. 1-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm127-1-1/