The Green-Tao theorem states that the primes contain arbitrarily long arithmetic progressions; its proof can be, somewhat inaccurately, broken up into the following two steps:
Thm1: Every constant-density subset of a pseudorandom set of integers contains arbitrarily long arithmetic progressions.
Thm2: The primes have constant density inside a pseudorandom set.
Of those, the main contribution of the paper is the first theorem, a “relative” version of Szemeredi’s theorem. In turn, its proof can be (even more inaccurately) broken up as
Thm 1.1: For every constant density subset D of a pseudorandom set there is a “model” set M that has constant density among the integers and is indistinguishable from D.
Thm 1.2 (Szemeredi) Every constant density subset of the integers contains arbitrarily long arithmetic progressions, and many of them.
Thm 1.3 A set with many long arithmetic progressions cannot be indistinguishable from a set with none.
Following this scheme is, of course, easier said than done. One wants to work with a definition of pseudorandomness that is weak enough that (2) is provable, but strong enough that the notion of indistinguishability implied by (1.1) is in turn strong enough that (1.3) holds. From now on I will focus on (1.1), which is a key step in the proof, though not the hardest.
Recently, Tao and Ziegler proved that the primes contain arbitrarily long “polynomial progressions” (progressions where the increments are given by polynomials rather than linear functions, as in the case of arithmetic progressions). Their paper contains a very clean formulation of (1.1), which I will now (accurately, this time) describe. (It is Theorem 7.1 in the paper. The language I use below is very different but equivalent.)
We fix a finite universe ; this could be in complexity-theoretic applications or in number-theoretic applications. Instead of working with subsets of , it will be more convenient to refer to probability distributions over ; if is a set, then is the uniform distribution over . We also fix a family of “easy” function . In a complexity-theoretic applications, this could be the set of boolean functions computed by circuits of bounded size. We think of two distributions as being -indistinguishable according to if for every function we have
and we think of a distribution as pseudorandom if it is indistinguishable from the uniform distribution . (This is all standard in cryptography and complexity theory.)
Now let’s define the natural analog of “dense subset” for distributions. We say that a distribution is -dense in if for every we have
Note that if and for some sets , then is -dense in if and only if and .
So we want to prove the following:
Theorem (Green, Tao, Ziegler)
Fix a family of tests and an ; then there is a “slightly larger” family and an such that if is an -pseudorandom distribution according to and is -dense in , then there is a distribution that is -dense in and that is -indistinguishable from according to .
[The reader may want to go back to (1.1) and check that this is a meaningful formalization of it, up to working with arbitrary distributions rather than sets. This is in fact the "inaccuracy" that I referred to above.]
In a complexity-theoretic setting, we would like to say that if is defined as all functions computable by circuits of size at most , then should be and should contain only functions computable by circuits of size . Unfortunately, if one follows the proof and makes some simplifications asuming contains only boolean functions, one sees that contains functions of the form , where , , and could be arbitrary and, in general, have circuit complexity exponential in and . Alternatively one may approximate as a low-degree polynomial and take the “most distinguishing monomial.” This will give a version of the Theorem (which leads to the actual statement of Thm 7.1 in the Tao-Ziegler paper) where contains only functions of the form , but then will be exponentially small in and . This means that one cannot apply the theorem to “cryptographically strong” notions of pseudorandomness and indistinguishability, and in general to any setting where and are super-logarithmic (not to mention super-linear).
This seems like an unavoidable consequence of the “finitary ergodic theoretic” technique of iterative partitioning and energy increment used in the proof, which always yields at least a singly exponential complexity.
Omer Reingold, Madhur Tulsiani, Salil Vadhan and I have recently come up with a different proof where both and the complexity of are polynomial. This gives, for example, a new characterization of the notion of pseudoentropy. Our proof is quite in the spirit of Nisan’s proof of Impagliazzo’s hard-core set theorem, and it is relatively simple. We can also deduce a version of the theorem where, as in Green-Tao-Ziegler, contains only bounded products of functions in . In doing so, however, we too incur an exponential loss, but the proof is somewhat simpler and demonstrates the applicability of complexity-theoretic techniques in arithmetic combinatorics.
Since we can use (ideas from) a proof of the hard core set theorem to prove the Green-Tao-Ziegler result, one may wonder whether one can use the “finitary ergodic theory” techniques of iterative partitioning and energy increment to prove the hard-core set theorem. Indeed, we do this too. In our proof, the reduction loses a factor that is exponential in certain parameters (while other proofs are polynomial), but one also gets a more “constructive” result.
If readers can stomach it, a forthcoming post will describe the complexity-theory-style proof of the Green-Tao-Ziegler result as well as the ergodic-theory-style proof of the Impagliazzo hard core set theorem.