Coherent systems over approximate lattices in amenable groups

Let $G$ be a second-countable amenable group with a uniform $k$-approximate lattice $\mathrm{\Lambda }$. For a projective discrete series representation $(\pi ,{\mathcal{H}}_{\pi })$ of $G$ of formal degree $d_{\pi }>0$, we show that $D^{-}(\mathrm{\Lambda })\ge d_{\pi }\mathrm{/}k$ is necessary for the coherent system $\pi (\mathrm{\Lambda })g$ to be complete in ${\mathcal{H}}_{\pi }$. In addition, we show that if $\pi ({\mathrm{\Lambda }}^2)g$ is minimal, then $D^{+}({\mathrm{\Lambda }}^2)\le d_{\pi }k$. Both necessary conditions recover sharp density theorems for uniform lattices and are new even for Gabor systems in $L^2(\mathbb{R})$. As an application of the approach, we also obtain necessary density conditions for coherent frames and Riesz sequences associated to general discrete sets. All results are valid for amenable unimodular groups of possibly exponential growth.