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.