  Let $G$ be a second-countable amenable group with a uniform $k$-approximate
lattice $\Lambda$. For a projective discrete series representation $(\pi,
\mathcal{H}_{\pi})$ of $G$ of formal degree $d_{\pi} > 0$, we show that
$D^-(\Lambda) \geq d_{\pi} / k$ is necessary for the coherent system
$\pi(\Lambda) g$ to be complete in $\mathcal{H}_{\pi}$. In addition, we show
that if $\pi(\Lambda^2) g$ is minimal, then $D^+ (\Lambda^2) \leq d_{\pi} k$.
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.
