# Goldbach-type theorems from dense models?

I'm not a number theorist, so apologies if this is trivial or obvious.

From what I understand of the results of Green-Tao-Ziegler on additive combinatorics in the primes, the main new technical tool is the "dense model theorem," which -- informally speaking -- is as follows:

If a set of integers $S \subset N$ is a dense subset of another "pseudorandom" set of integers, then there's another set of integers $S' \subset N$ such that $S'$ has positive density in the integers and $S, S'$ are "indistinguishable" by a certain class of test functions.

They then use some work of Goldston and Yildirim to show that the primes satisfy the given hypothesis, and note that if the primes failed to contain long arithmetic/polynomial progressions and $S'$ did, they'd be distinguishable by the class of functions. Applying Szemeredi's theorem, the proof is complete.

Obviously I'm skimming over a great deal of technical detail, but I'm led to believe that this is a reasonably accurate high-level view of the basic approach.

My question(s), then: Can one use a similar approach to obtain "Goldbach-type" results, stating that every sufficiently large integer is the sum of at most k primes? Is this already implicit in the Goldston-Yildirim "black box?" If we can't get Goldbach-type theorems by using dense models, what's the central obstacle to doing so?

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