**Notation:**


We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.


**Question:**


Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M\_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$ with the following property?


Suppose the foliation $M\_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M\_{\alpha} \cap P$ is dense in $M\_{\alpha}$ is dense in $(0, 1)$.



Existence of a certain foliation of $\mathbb R^n$
**EDIT:** Originally I could prove that there is such a foliation by topological manifolds:


Clearly, if $\mathbb{Q}^n$ is the set if points with all rational coordinates, you can have a foliation by parallel hyperplanes $H\_\alpha$. Now, for any set $P$ there that is countable and dense there is homepmorphism $\Phi:\mathbb{R}^n\to\mathbb{R}^n$ such that $\Phi(\mathbb{Q}^n)=P$, see Claim A) on page 44 in [1]. Then $M\_\alpha=\Phi(H\_\alpha)$ is a foliation by topological manifolds.


However, as pointed out by Alessandro Codenotti in a comment below, Morayne proved in [2] that there is $\Phi$ with the properties listed above that is a volume preserving analytic diffeomorphism so in fact the foliation can be done by analytic manifolds.


**[1] W. Hurewicz, H. Wallman,** *Dimension Theory*.
Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton, N. J., 1941.


**[2] M. Morayne,**
Measure preserving analytic diffeomorphisms of countable dense sets in $C^n$ and $R^n$. *Colloq. Math.* 52 (1987), 93–98.


