While doing my researches, I encountered the following problem.
Let $f:[0,1]^n\rightarrow \mathbb{R}^{n+k}$ be an arbitrary continuous function.
I want to make this function an embedding by perturbing it, and the perturbation function should be continuous and with additional coordinates: if possible I want to be able to choose its structure, particularly the additional coordinates.
The problem can be formulated as follows.
Let $p_{d_1,d_2}:\mathbb{R}^{d_1}\rightarrow \mathbb{R}^{d_2}$ be a projection to the first $d_2$ coordinates.
Then, a number $l$ may satisfy the following condition:
for a continuous function $f:[0,1]^n\rightarrow \mathbb{R}^{n+k}$ and a positive value $\epsilon$, there exists an embedding $F:\mathbb{R}^n\rightarrow \mathbb{R}^{n+k+l}$ such that:
$$ \|f-p_{n+k+l, n+k} \circ F\|_{\infty}<\epsilon.$$
The minimum value $l_{min}$ of $l$ obviously depends on $n$ and $k$ and I want to determine it.
I have found that this problem shares similarities with the Whitney Embedding Theorem.
If $n+k \geq 2n+1$, then it is possible to achieve injectivity solely through perturbation, without the need for additional dimensions.
In other words, if $k>n$, then $l_{\min}=0$ according to Theorem 2 of [1] or Lemma 3.18 of [2].
However, I have not been able to find any clues for smaller values of $k$.
Are there any tools or references available for dealing with this type of problem?
Any assistance would be greatly appreciated.
References
[1] Hassler Whitney, "Differentiable manifolds", Annals of Mathematics, pp. 645–680, 1936, JFM 62.1454.01, Zbl 0015.32001
[2] Milton Persson, The Whitney embedding theorem, Institutionen för matematik och matematisk statistik, VT 2014.
Edit:
I have found a reference [3].
I am considering the case $\frac{n}{2}<k<n$ where the self-intersection is at most 2-fold.
Consider an immersion $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n+k}$.
Let $N$ be a connected component of the intersection of the image.
Then, $f^{-1}(N)$ can have one or two connected components.
If there are two components, then $f: f^{-1}(N) \rightarrow N$ is trivial double covering which means that by adding an additional function that has value $0$ on one component and $1$ on another, we can extend $f$ to an injective function. It implies that $l=1$.
A more complicated case is that $f^{-1}(N)$ itself is connected.
I think that the worst case is something like $f^{-1}(N) \approx S^{n-k}$ and $N\approx \mathbb{RP}^{n-k}$.
In this case, by Borsuk-Ulam theorem, for any additional function $g:f^{-1}(N)\rightarrow \mathbb{R}^{n-k}$, there exists antipodal points $x$ and $-x$ such that $g(x)= g(-x)$.
In this case, we may get $l> n-k$ and $n+k+l >2n$.
Therefore, my question is reduced to whether we can get a 2-fold intersection that the covering from $f^{-1}(N)$ to $N$ is hard to disentangle.(Something like $f^{-1}(N) \approx S^{n-k}$ and $N\approx \mathbb{RP}^{n-k}$. )
But I haven't found an example of such cases, and Theorem 3.1 of [3] says that $N$ is orientable if $k$ is even (So obviously not $\mathbb{RP}^n$).
Is anyone who can give some concrete examples of such cases or any other hints?
[3] Lashof, R. K., and S. Smale. "Self-intersections of immersed manifolds." Journal of Mathematics and Mechanics (1959): 143-157.
