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2012 | 127 | 1 | 1-15
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Existence of large ε-Kronecker and FZI₀(U) sets in discrete abelian groups

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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 → 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.
Słowa kluczowe
  • Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
  • P.O. Box 2031 Haines Junction, YT, Canada Y0B 1L0
  • Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1
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