In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked

Is it true, as rumours have it, that you started to work on the embedding problem as a result of a bet?

Nash answered

I began to work on it. Then I got shifted onto the $𝐶^1$ case. It turned out that one could do it in this case with very few excess dimensions of the embedding space compared with the manifold. I did it with two but then Kuiper did it with only one. But he did not do it smoothly, which seemed to be the right thing—since you are given something smooth, it should have a smooth answer.

and

But a few years later, I made the generalisation to smooth. I published it in a paper with four parts. There is an error, I can confess now. Some forty years after the paper was published, the logician Robert M. Solovay from the University of California sent me a communication pointing out the error. I thought: “How could it be?” I started to look at it and finally I realized the error in that if you want to do a smooth embedding and you have an infinite manifold, you divide it up into portions and you have embeddings for a certain amount of metric on each portion. So you are dividing it up into a number of things: smaller, finite manifolds. But what I had done was a failure in logic. I had proved that—how can I express it?—that points local enough to any point where it was spread out and differentiated perfectly if you take points close enough to one point; but for two different points it could happen that they were mapped onto the same point.

My question is:

1- What did Nash mean by very few excess dimensions of the embedding space compared with the manifold? It means that the they can do it in the case the dimension of the embedding space is a little bit greater than the dimension of the manifold, doesn't it?

2- What did Nash mean by his generalization to smooth?

3- What did Nash mean by a certain amout of metric on each portion? Does this mean that each portion has some different metrics?

4- What did Nash mean by "I had proved that that points local enough to any point where it was spread out and differentiated perfectly if you take points close enough to one point"? How can a point spread out and differentiated?

Well, I don't want to make a cross post but this question were posted two days on MSE but there is no answer so I decide to post it here.

Please explain for me.

Thanks.