$\varepsilon $-Kronecker and $I_{0}$ sets in abelian groups, III: interpolation by measures on small sets

Volume 171 / 2005

Colin C. Graham, Kathryn E. Hare Studia Mathematica 171 (2005), 15-32 MSC: Primary 42A55, 43A46; Secondary 43A05, 43A25. DOI: 10.4064/sm171-1-2


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.


  • Colin C. GrahamUniversity of British Columbia
  • Kathryn E. HareDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, ON, N2L 3G1 Canada

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