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Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply connected and has non-degenerate intersection form. Suppose we 'approximate' the given manifold by a sequence of compact subsets of $\mathbb{R}^{m}$ such that:

$N_{i}\subset M,\;\; |\dim(N_{i})-4|<\frac{1}{i},\;\; \mu(M-N_{i})<\epsilon_{i},\;\; \epsilon_{i}\rightarrow 0$ Here we are using Hausdorff dimension.

I asked an Geometric Analysis Professor on this problem, and he said the limit of $N_{i}$ might not tell us anything about $M$, especially $N_{i}$ has far worse analytical properties than $M$. So my question is - can we put a nice differentiable structure on $N_{i}$ such that the above limit makes some sense? For example, the unit real line may be 'approximated' by various fat Cantor sets. This question is quite unclear to me, so I wish to discuss at here.

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    I'm not entirely sure what you are going for, but I think that the answer is "very, very difficult." Strichartz has [a book](https://books.google.com/books?id=CNWJRvXAdhAC&printsec=frontcover&dq=strichartz+analysis+on+fractals&hl=en&sa=X&ved=0ahUKEwiZoeT04ZHWAhVJ1WMKHWPnAIsQ6AEIJzAA#v=onepage&q=strichartz%20analysis%20on%20fractals&f=false) on the topic (or, at least, on a related topic).2017-09-06

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