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·Δ is I₀ for all finite subsets Δ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-1-2, author = {Colin C. Graham and Kathryn E. Hare}, title = {e-Kronecker and I0 sets in abelian groups, III: interpolation by measures on small sets}, journal = {Studia Mathematica}, volume = {166}, year = {2005}, pages = {15-32}, zbl = {1086.43005}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-1-2} }
Colin C. Graham; Kathryn E. Hare. ε-Kronecker and I₀ sets in abelian groups, III: interpolation by measures on small sets. Studia Mathematica, Tome 166 (2005) pp. 15-32. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-1-2/