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

Abstract: 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.