Let $f: M \rightarrow N$ be a smooth map between two submanifolds of $\mathbb{R}^{m}$, $\mathbb{R}^{n}$ respectively. Sard's famous theorem asserts that the set of critical values $C$ of $f$ has measure zero.
My question is: Does every null set in $\mathbb{R}^n$ arise as the set of critical values of some smooth map $f$ as above?
As a start: For $n=1$ and $C \subset \mathbb{R}$ countable, I think one can construct such a map. Namely, let $M=\coprod_{c \in C} \mathbb{R}$ be the disjoint union of $|C|$ copies of $\mathbb{R}$ and $f: M \rightarrow \mathbb{R}$ be defined by $(x,c) \mapsto x^2+c$. $M$ is a one-dimensional real manifold (it is $2$nd countable, Hausdorff and carries a natural smooth structure coming from the one on $\mathbb{R}$). Hence, by Whitney, it can be embedded as a submanifold of some $\mathbb{R}^m$. Moreover, the set of critical values of $f$ is exactly $C$.
But this approach does not seem to work in general. If $C \subset \mathbb{R}$ is an uncountable null set, for example, then $M=\coprod_{c \in C} \mathbb{R}$ is not $2$nd countable, hence is not a smooth manifold in the usual sense and Whitney's theorem does not apply.
Any help towards an answer to my question is much appreciated! In particular, references are also welcome.