Riesz bases of exponentials and the Bohr topology

We provide a necessary and sufficient condition to ensure that a multi-tile $\mathrm{\Omega }$ of $R^d$ of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for $L^2(\mathrm{\Omega })$. New examples are given and this characterization is generalized to abstract locally compact abelian groups.