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?


Yes, this can be done, provided that k is at least 3. A typical example is given in this paper: http://arxiv.org/abs/math/0701240 . (This uses a slightly older Fourier-based method of Ben that predates his work with me, but is definitely in the same spirit - Ben's paper was very inspirational for our joint work.)

For k=2, unfortunately, the type of indistinguishability offered by the dense model theorem is not strong enough to say anything. The k=2 problem is very similar to the twin prime conjecture (in both cases, one is trying to seek an additive pattern in the primes with only one degree of freedom). If one deletes all the twin primes from the set of primes, one gets a new set which is indistinguishable from the set of primes in the sense of the dense model theorem (because the twin primes have density zero inside the primes, by Brun's theorem), but of course the latter set has no more twins. One can pull off a similar trick with representations of N as the sum of two primes. But one can't do it with sums of three or more primes - there are too many representations to delete them all just by removing a few primes.


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