After having discussed the notion of Gowers uniformity, the statement of Szemeredi’s theorem, and the proof for progressions of length 3, let me try to say something about the case of progressions of length 4 or more, and the several related open questions.

As already discussed, Gowers’s proof is based on an “algorithm” that, given a subset A of ZN of size dN, does one of the following two things:

1. It immediately finds a progression of length k in A; or
2. it constructs a subset A’ of ZN’ such that: (i) if there is a progression of length k in A’, then there is also such a progression in A; (ii) A’ has size at least (d+poly(d))*N’; N’ is at least a constant root of N.

As in the proof for the case of progressions of length 3, once we have such an algorithm we are able to find a progression of length k in A, provided that, initially, N is at least exp(exp(poly(1/d))).

How does the “algorithm” work? Let us, again, identify the set A with its characteristic function A: ZN -> {0,1}, and consider the expression

(*) Ex,y A(x)*A(x+y)*A(x+2y)*…*A(x+(k-1)*y)

It is possible to show that (*) equals dk (which is what we would expect if A were a “pseudorandom” set) plus or minus an error term that depends on Uk-1(A-d), where Uk-1 is the dimension-(k-1) Gowers uniformity as defined earlier. A result of this kind (relating Gowers uniformity and number of arithmetic progressions) is called a “generalized Von Neumann theorem” in the literature, for reasons that I unfortunately ignore. The proof is not difficult at all, it is just a series of applications of Cauchy-Schwartz. The genius is in the choice of the right definition. (And the technical difficulties are in what comes next.)

This means that if the dimension-(k-1) Gowers uniformity of the function f(x):=A(x)-d is small, then A contains a lot of length-k arithmetic progressions, and we are in case (1)

What remains to prove is that if the set A is such that f(x):=A(x)-d has noticeably large dimension-(k-1) Gowers uniformity then we can find a set A’ in ZN’ as in case (2) above. Gowers’s paper is approximately 130 pages long, and about 100 pages are devoted to proving just that.

Obviously, I have no idea what goes on in those 100 pages, but (based on later work by Green and Tao) I can guess how it would be if we were trying to prove Szemeredi’s theorem not in ZN but in Fnp with p prime and larger than k. Translated to such a setting (I think), Gowers’s argument would proceed by showing that if Uk(A-d) is at least eps, then there is an affince subspace V of Fnp of dimension n/O(1) such that A restricted to V has density at least d+poly(eps). This, in turn, would be proved by showing

1. If f has noticeably large dimension-(k-1) Gowers uniformity, then there is an affine subspace V of dimension n/O(1) and a polynomial phase function g of degree (k-2) such that, restricted to V, f and g are noticeably correlated
2. If f is a function that has noticeable correlation with a low degree polynomial phase function, then there is a subspace of dimension n/O(1) such that f is correlated to a linear function when restricted to that subspace.

Let’s make things simple by restricting ourselves to Boolean function (even though this p=2 case has no direct application to results about arithmetic progressions).

Then, part (1) is saying that if f:{0,1}n->{-1,1} has noticeably large dimension-k Gowers uniformity, then there is a polynomial p() of degree (k-1) over Z2, and an affine subspace V of dimesion n/O(1), such that f(x) and (-1)p(x) agree on noticeably more than 1/2 of the inputs x. Part (2) is saying that if f:{0,1}n->{-1,1} and (-1)p(x) agree on noticeably more than 1/2 of the inputs, where p is a low-degree polynomial, then there is an affine subspace V of dimension n/O(1) such that, restricted to V, f has agreement with a linear function on noticeably more than 1/2 of the inputs.

The two results together imply that if f has noticeably large Gowers uniformity then there is a large sub-space in which f is correlated with a linear function. (To reach the conclusion, you need to apply part (1), then do a change of variables so that now the V of part (1) looks like {0,1}n/O(1), and then apply part(2).)

Actually, what I just wrote may be an open question, but it should be provable along the lines of Gowers’s proof, and it should be much easier to prove.

For the case of progressions of length 4 in Fnp, for small prime p>4, Green and Tao prove that if f is a bounded function of noticeably large dimension-3 Gowers uniformity, then f is correlated to a degree-2 polynomial over all of Fnp, and the correlation is especially good on a subspace of dimension n-O(1). (As opposed to n/O(1).) This improvement, and several additional ideas, lead to an improved quantitative version of Szemeredi’s theorem for progressions of length 4 in Fnp, and they also announced a similar improved result for progression of length 4 over the integers.

Their proof relies on the analysis of a certain “linearity test in the highly noisy case,” which is also used (and, I think, proved for the first time) in Gowers’s paper. In the, simpler, Boolean, case, the result is

Let F:{0,1}n -> {0,1}m, and suppose that

Prx,y[F(x)+F(y) = F(x+y)] > eps

where all operations are bitwise mod 2. Then there is an eps’ (depending only on eps, but independent of n,m) and a matrix M such that the functions x->F(x) and x->Mx have agreement at least eps’

(Update: as Alex explains in his comment below, the above sentence is quite misleading. Gowers proves quite a different statement in ZN, a setting in which an analogous “highly noise linearity test” statement is provably false, and where it is difficult to even get the statement of the right analog. The Boolean statement above is proved for the first time in Alex’s paper, with a proof whose outline is similar to Gowers’s; Green and Tao prove the highly noise linearity test result in other prime fields, also following Gowers’s argument.)

In the known proof, which uses difficult results from additive combinatorics, eps’ is exponentially small in eps. Here two major open questions are: (i) find a simple proof and (ii) find a proof where eps’=poly(eps)

More ambitiously, there could be a simple proof that (say, in the Boolean case) if f has noticeable Gowers uniformity, then f is noticeably correlated with a linear function when restricted to a large subspace. The experts in the area would probably be able to turn a proof in the Boolean case to a proof that applies to the setting where we have ZN instead of {0,1}n, Bohr sets instead of subspaces, and things I do not understand instead of polynomials. Even so, if the proof in Boolean setting were simple enough, the whole thing could be significantly simpler than Gowers’s proof, and there could be room for quantitative improvements.