$\varepsilon $-Kronecker and $I_{0}$ sets in abelian groups, III: interpolation by measures on small sets
Volume 171 / 2005
Abstract
Let $U$ be an open subset of a locally compact abelian group $G$ and let $E$ be a subset of its dual group ${\mit\Gamma} $. We say $E$ is $I_0(U)$ 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\cdot {\mit\Delta} $ is $I_0$ for all finite subsets ${\mit\Delta} $ of a torsion-free ${\mit\Gamma} $, then for each open $U\subset G$ there exists a finite set $F\subset E$ such that $E\setminus F$ is $I_0(U)$. When $G$ is connected, $F$ can be taken to be empty. We obtain a much stronger form of that for Hadamard sets and $\varepsilon $-Kronecker sets, and a slightly weaker general form when ${\mit\Gamma} $ has torsion. This extends previously known results for Sidon, $\varepsilon $-Kronecker, and Hadamard sets.
