  We show that on any smooth compact connected manifold of dimension $m\geq 2$
admitting a smooth non-trivial circle action $\mathcal{S} =
\left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing
$C^{\infty}$-diffeomorphisms which preserve both a smooth volume $\nu$ and a
measurable Riemannian metric is dense in $\mathcal{A}_{\alpha} \left(M\right)=
\overline{ \left\{h \circ S_{\alpha} \circ h^{-1} : h \in
\text{Diff}^{\infty}\left(M, \nu\right) \right\}}^{C^{\infty}}$ for every
Liouvillean number $\alpha$. The proof is based on a quantitative version of
the Anosov-Katok-method with explicitly constructed conjugation maps and
partitions.
