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Ben Green, Tamar Ziegler and I have just uploaded to the arXiv our paper “An inverse theorem for the Gowers norm“. This paper establishes the next case of the inverse conjecture for the Gowers norm for the integers (after the case, which was done by Ben and myself a few years ago). This conjecture has a number of combinatorial and number-theoretic consequences, for instance by combining this new inverse theorem with previous results, one can now get the correct asymptotic for the number of arithmetic progressions of primes of length five in any large interval .

To state the inverse conjecture properly requires a certain amount of notation. Given a function and a shift , define the multiplicative derivative

and then define the Gowers norm of a function to (essentially) be the quantity

where we extend f by zero outside of . (Actually, we use a slightly different normalisation to ensure that the function 1 has a norm of 1, but never mind this for now.)

Informally, the Gowers norm measures the amount of bias present in the multiplicative derivatives of . In particular, if for some polynomial , then the derivative of is identically 1, and so is the Gowers norm.

However, polynomial phases are not the only functions with large Gowers norm. For instance, consider the function , which is what we call a *quadratic bracket polynomial phase*. This function isn’t quite quadratic, but it is close enough to being quadratic (because one has the approximate linearity relationship holding a good fraction of the time) that it turns out that third derivative is trivial fairly often, and the Gowers norm is comparable to 1. This bracket polynomial phase can be modeled as a *nilsequence* , where is a polynomial orbit on a nilmanifold , which in this case has step 2. (The function F is only piecewise smooth, due to the discontinuity in the floor function , so strictly speaking we would classify this as an *almost nilsequence* rather than a nilsequence, but let us ignore this technical issue here.) In fact, there is a very close relationship between nilsequences and bracket polynomial phases, but I will detail this in a later post.

The inverse conjecture for the Gowers norm, GI(s), asserts that such nilsequences are the only obstruction to the Gowers norm being small. Roughly speaking, it goes like this:

Inverse conjecture, GI(s).(Informal statement) Suppose that is bounded but has large norm. Then there is an s-step nilsequence of “bounded complexity” that correlates with f.

This conjecture is trivial for s=0, is a short consequence of Fourier analysis when s=1, and was proven for s=2 by Ben and myself. In this paper we establish the s=3 case. An equivalent formulation in this case is that any bounded function of large norm must correlate with a “bracket cubic phase”, which is the product of a bounded number of phases from the following list

(*)

for various real numbers .

It appears that our methods also work in higher step, though for technical reasons it is convenient to make a number of adjustments to our arguments to do so, most notably a switch from standard analysis to non-standard analysis, about which I hope to say more later. But there are a number of simplifications available on the s=3 case which make the argument significantly shorter, and so we will be writing the higher s argument in a separate paper.

The arguments largely follow those for the s=2 case (which in turn are based on this paper of Gowers). Two major new ingredients are a deployment of a normal form and equidistribution theory for bracket quadratic phases, and a combinatorial decomposition of frequency space which we call the sunflower decomposition. I will sketch these ideas below the fold.

Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our paper “An inverse theorem for the uniformity seminorms associated with the action of “. This paper establishes the ergodic inverse theorems that are needed in our other recent paper to establish the inverse conjecture for the Gowers norms over finite fields in high characteristic (and to establish a partial result in low characteristic), as follows:

Theorem.Let be a finite field of characteristic p. Suppose that is a probability space with an ergodic measure-preserving action of . Let be such that the Gowers-Host-Kra seminorm (defined in a previous post) is non-zero.

- In the high-characteristic case , there exists a phase polynomial g of degree <k (as defined in the previous post) such that .
- In general characteristic, there exists a phase polynomial of degree <C(k) for some C(k) depending only on k such that .

This theorem is closely analogous to a similar theorem of Host and Kra on ergodic actions of , in which the role of phase polynomials is played by functions that arise from nilsystem factors of X. Indeed, our arguments rely heavily on the machinery of Host and Kra.

The paper is rather technical (60+ pages!) and difficult to describe in detail here, but I will try to sketch out (in very broad brush strokes) what the key steps in the proof of part 2 of the theorem are. (Part 1 is similar but requires a more delicate analysis at various stages, keeping more careful track of the degrees of various polynomials.)

Let be an integer. The concept of a polynomial of one variable of degree (or ) can be defined in one of two equivalent ways:

- (Global definition) is a polynomial of degree iff it can be written in the form for some coefficients .
- (Local definition) is a polynomial of degree if it is k-times continuously differentiable and .

From single variable calculus we know that if P is a polynomial in the global sense, then it is a polynomial in the local sense; conversely, if P is a polynomial in the local sense, then from the Taylor series expansion

we see that P is a polynomial in the global sense. We make the trivial remark that we have no difficulty dividing by here, because the field is of characteristic zero.

The above equivalence carries over to higher dimensions:

- (Global definition) is a polynomial of degree iff it can be written in the form for some coefficients .
- (Local definition) is a polynomial of degree if it is k-times continuously differentiable and for all .

Again, it is not difficult to use several variable calculus to show that these two definitions of a polynomial are equivalent.

The purpose of this (somewhat technical) post here is to record some basic analogues of the above facts in finite characteristic, in which the underlying domain of the polynomial P is F or for some finite field F. In the “classical” case when the range of P is also the field F, it is a well-known fact (which we reproduce here) that the local and global definitions of polynomial are equivalent. But in the “non-classical” case, when P ranges in a more general group (and in particular in the unit circle ), the global definition needs to be corrected somewhat by adding some new monomials to the classical ones . Once one does this, one can recover the equivalence between the local and global definitions.

(The results here are derived from forthcoming work with Vitaly Bergelson and Tamar Ziegler.)

Tamar Ziegler and I have just uploaded to the arXiv our paper, “The inverse conjecture for the Gowers norm over finite fields via the correspondence principle“, submitted to Analysis & PDE. As announced a few months ago in this blog post, this paper establishes (most of) the inverse conjecture for the Gowers norm from an ergodic theory analogue of this conjecture (in a forthcoming paper by Vitaly Bergelson, Tamar Ziegler, and myself, which should be ready shortly), using a variant of the Furstenberg correspondence principle. Our papers were held up for a while due to some unexpected technical difficulties arising in the low characteristic case; as a consequence, our paper only establishes the full inverse conjecture in the high characteristic case , and gives a partial result in the low characteristic case .

In the rest of this post, I would like to describe the inverse conjecture (in both combinatorial and ergodic forms), and sketch how one deduces one from the other via the correspondence principle (together with two additional ingredients, namely a statistical sampling lemma and a local testability result for polynomials).

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