$\varepsilon $-Kronecker and $I_{0}$ sets in abelian groups, IV: interpolation by non-negative measures
Volume 177 / 2006
Studia Mathematica 177 (2006), 9-24
MSC: Primary 42A55, 43A46; Secondary 43A05, 43A25, 42A82.
DOI: 10.4064/sm177-1-2
Abstract
A subset $E$ of a discrete abelian group is a “Fatou–Zygmund interpolation set” ($F\kern-.75pt ZI_0$ 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 $F\kern-.75pt ZI_0$ set of the same cardinality (if the group is torsion free, a stronger interpolation property holds) and that $\varepsilon $-Kronecker sets are $F\kern-.75pt ZI_0$ (with that stronger interpolation property).
