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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 ameasurable 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 everyLiouvillean number $\alpha$. The proof is based on a quantitative version ofthe Anosov-Katok-method with explicitly constructed conjugation maps andpartitions.

Weakly mixing diffeomorphisms preserving a measurable Riemannian metric  are dense in $\mathcal{A}_{\alpha}\left(M\right)$ for arbitrary Liouvillean  number $\alpha$