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Let U be an open subset of a locally compact abelian group G and let E be a subset of its dual group Γ . We say E is I0(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 · ∆ is I0 for all finite subsets ∆ of a torsion-free Γ , then for each open U ⊂ G there exists a finite set F ⊂ E such that E \ F is I0(U). When G is connected, F can be taken to be empty. We obtain a much stronger form of that for Hadamard sets and ε-Kronecker sets, and a slightly weaker general form when Γ has torsion. This extends previously known results for Sidon, ε-Kronecker, and Hadamard sets.

ε-Kronecker and I0sets in abelian groups, III: interpolation by measures on small sets

ε-Kronecker and I₀sets in abelian groups, III: interpolation by measures on small sets