As mentioned in this question (https://math.stackexchange.com/questions/416607/show-that-fc-has-hausdorff-dimension-at-most-zero/446049#446049), in 1965 Sard proved the following result (paraphrased slightly here):
Define the function $c : \mathbb{N}^{\geq 0} \times \mathbb{R}^{>0} \rightarrow \mathbb{R}$ via: $$c(k,\rho ) := \begin{cases} 1 &\text{ if }\ k<\rho \\ 2+\frac{k(k-\rho)}{\rho} &\text{ if }\ 0<\rho\leq k \end{cases}$$ Further, whenever $f : \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a $C^q$ map, define:
$A_r := \{x \in \mathbb{R}^d : \text{rank}(Df(x)) \leq r\}$ for each $r \in \{0,...,d\}$.
Theorem. If $\rho \in \mathbb{R}^{>0}$ and $q \geq c(d-r, \rho)$. Then: $$\mathcal{H}^{r + \rho}(f[A_r]) = 0$$
By setting $r = 0$ and $\rho = d - \varepsilon$, this tells us that $\mathcal{H}^{d - \varepsilon}(f[A_0]) = 0$ as long as $q \geq 2 + \frac{d \varepsilon}{d - \varepsilon}$.
Now as long as $\varepsilon \leq \frac{1}{2}$, the above inequality will always be satisfied for $C^3$ functions. But what about $C^1$ and $C^2$ functions? Is it possible for a $C^1$ or $C^2$ function to have $f[A_0]$ be Hausdorff $d$-dimensional for example?
Also Sard's theorem is additive, discussing the nullity with respect to $\mathcal{H}^{r + \rho}$. Are there any analogous results which are multiplicative?
For example does there exists a function $c^*$ such that whenever $q \geq c^*(\frac{d}{r}, \rho)$ we have:
$$\mathcal{H}^{\rho r}(f[A_r]) = 0$$
