Green, Tao and Ziegler, in their works on patterns in the primes, prove a general result of the following form: if is a set, is a, possibly very sparse, “pseudorandom” susbset of , and is a dense subset of , then may be “modeled” by a large set which has the same density in as the density of in .
They use this result with being the integers in a large interval , being the “almost-primes” in (integers with no small factor), and being the primes in . Since the almost-primes can be proved to be “pseudorandom” in a fairly strong sense, and since the density of the primes in the almost-primes is at least an absolute constant, it follows that the primes are “indistinguishable” from a large set containing a constant fraction of all integers. Since such large sets are known to contain arbitrarily long arithmetic progressions, as proved by Szemeredi, Green and Tao are able to prove that the primes too must contain arbitrarily long arithmetic progressions. Such large sets are also known to contain arbitrarily long “polynomial progressions,” as proved by Bergelson and Leibman, and this allows Tao and Ziegler to argue that the primes too much contain arbitrarily long polynomial progressions.
(The above account is not completely accurate, but it is not lying too much.)
As announced last October here, and here, Omer Reingold, Madhur Tulsiani, Salil Vadhan and I found a new proof of this “dense model” theorem, which uses the min-max theorem of game theory (or, depending on the language that you prefer to use, the duality of linear programming or the Hahn Banach theorem) and was inspired by Nisan’s proof of the Impagliazzo hard-core set theorem. In complexity-theoretic applications of the theorem, our reduction has polynomial complexity, while the previous work incurred an exponential loss.
After long procrastination, we recently wrote up a paper about these results.
In the Fall, we received some feedback from additive combinatorialists that while our proof of the Green-Tao-Ziegler result was technically simpler than the original one, the language we used was hard to follow. (That’s easy to believe, because it took us a while to understand the language in which the original proof was written.) We then wrote an expository note of the proof in the analyst’s language. When we were about to release the paper and the note, we were contacted by Tim Gowers, who, last Summer, had independently discovered a proof of the Green-Tao-Ziegler results via the Hahn-Banach theorem, essentially with the same argument. (He also found other applications of the technique in additive combinatorics. The issue of polynomial complexity, which does not arise in his applications, is not considered.)
Gowers announced his results in April at a talk at the Fields institute in Toronto. (Audio and slides are available online.)
Gowers’ paper already contains the proof presented in the “analyst language,” making our expository note not so useful any more; we have still posted it anyways because, by explaining how one translates from one notation to the other, it can be a short starting point for the computer scientist who is interested in trying to read Gowers’ paper, or for the combinatorialist who is interested in trying to read our paper.