The failure of spectral synthesis on some types of discrete abelian groups (Q1433377)

The setting is the space \({\mathcal C}(G)\) of complex-valued functions on a discrete Abelian group \(G\). The topology is that of pointwise convergence. The question of synthesis is ``Does every non-zero closed translation invariant subspace contain sufficient exponential monomials for their linear hull to be dense?'' (An exponential monomial is the product of a homomorphism into the multiplicative group of non-zero complex numbers by a homomorphism into the additive group of complex numbers.) The question was thought to have been settled in the affirmative for arbitrary discrete abelian groups by \textit{R. J. Elliot} [Proc. Camb. Philos. Soc. 61, 617--620 (1965; Zbl 0138.07402)], but the author of the present paper shows that synthesis fails on any Abelian group with torsion-free rank at least \(\omega\).