Smoothing a periodic function of two variables

Question

Let $F \colon \mathbb{R}^2 \to \mathbb{R}^n$ be a $C^{1}$-function 1-periodic in each variable, so it can be considered as a function on the flat torus $\mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$. We say that a point $\theta \in \mathbb{T}^2$ is a critical point for $F$ if rank of the differential of $F$ at the point $\theta$ is degenerate, i. e. $$\operatorname{rank}\left[ \frac{\partial F}{\partial \theta}(\theta) \right]<2.$$

My question is. Is there for any given $\varepsilon>0$ and natural $k$ a $C^1$-smooth 1-periodic function $\widetilde{F}$ such that $\|F-\widetilde{F}\|_{C^1}<\varepsilon$ and measure (the Lebesgue measure on $\mathbb{T}^2$) of the set of critical points of $\widetilde{F}$ is less than $\frac{1}{k}$.

Answer 1

Let's suppose that $n \geq 2$ and take a $C^{1}$-function $H \colon \mathbb{R}^2 \to \mathbb{R}^n$ such that the set of its critical points has measure zero. Consider the family of functions $\Phi_{\alpha} = F + \alpha H$ for $\alpha \in \mathbb{R}$. It turns out that there exists a sequence $\alpha_{n}$ tending to zero and the set of critical points of $\Phi_{\alpha_n}$ has measure zero. To see this one has to compare two facts.

1. Consider a two-dimensional subspace $L \subset \mathcal{L}(E,F)$ in the space of all linear operators from $E$ to $F$. If there are three pairwise non-proportional operators $A,B,C$ (i. e. every two of them is a basis for $L$) with ranges of dimension $1$ then each non-zero operator from $L$ has range of dimension $1$.

2. A family of measurable sets (in the space of finite measure) of positive measure such that the intersection of any three of them has measure zero is at most countable.