Dichotomy Paper

I have a couple questions about the Dichotomy paper of Miller:

1) In Theorem 9 (the Perfect Set Theorem) there is a step that says to “recursively construct a continuous \psi:2^{\omega}\to 2^{\omega} such that (\varphi\circ\psi)[N_s]\cap(\varphi\circ\psi)[N_t]=\emptyset whenever t,s\in 2^n for some n. He says that you can do it because varphi|U is non-constant for nonempty open sets.

I’ve tried doing something like the schemes where we assign a clopen set to each node of 2^{<\omega} and assign to an x\in 2^{\omega} the intersection of the clopen sets for its inital segments. But I haven't really been able to make it work.

2) In the proof of the silver Dichotomy, we get a perfect clique and the homomorphism ends these to perfectly many equivalence classes. But I am unsure of why there must be a perfect clique.

3) Finally, for Lusin-Novikov, I'm not entirely sure why G_0 has to map into one vertical section. I know that each connected component has to go into one section, since we are applying a homomorphism. But the graph isn't connected (right?).

Help with these would be appreciated. Overall the paper looks interesting. I haven't been able to get into the rest of it as much I would have liked. But I think working through these examples will help when it comes to understanding the rest of the paper.


Image of Fourier Transform

In analysis today, we were talking about the fourier transform and how as a map from L_2 to L_2 it is onto, but as a map from L_1 to C_0 it isn’t onto and the range isn’t even closed. Then Dr. Jury said that the range is really hard to describe.

So after class I asked him if “hard to describe” meant “analytic but non-Borel” since the range is analytic by definition. He didn’t know off-hand. I did a quick look online and didn’t find other than people saying that the range is hard to characterize.

So, is there any result saying it is \Sigma^1_1 complete or at least non-Borel? I would be interested to know, and to see the proof.