The redundancy of Gabor type unitary systems on locally compact abelian groups

In this paper, we extend the redundancy theorem of J. Gabardo and D. Han for Gabor type unitary systems indexed by full-rank lattices in Euclidean spaces to the setting of locally compact abelian (LCA) groups. Let $G$ be an LCA group, let $\Lambda$ be a uniform lattice in $G$, let $\alpha$ be an automorphism of $G$, and let $\beta$ be an automorphism of $\hat{G}$. We show that the redundancy of a Gabor type unitary system indexed by $\alpha \left(\Lambda \right)\times \beta \left({\Lambda }^{\perp }\right)$ equals the reciprocal of the density of this index set. As an application, we give a new proof of the famous time-frequency density theorem in Gabor analysis.