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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 $F^\infty_p$“. 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 ${\Bbb F}$ be a finite field of characteristic p.  Suppose that $X = (X,{\mathcal B},\mu)$ is a probability space with an ergodic measure-preserving action $(T_g)_{g \in {\Bbb F}^\omega}$ of ${\Bbb F}^\omega$.  Let $f \in L^\infty(X)$ be such that the Gowers-Host-Kra seminorm $\|f\|_{U^k(X)}$ (defined in a previous post) is non-zero.

1. In the high-characteristic case $p \geq k$, there exists a phase polynomial g of degree <k (as defined in the previous post) such that $|\int_X f \overline{g}\ d\mu| > 0$.
2. In general characteristic, there exists a phase polynomial of degree <C(k) for some C(k) depending only on k such that $|\int_X f \overline{g}\ d\mu| > 0$.

This theorem is closely analogous to a similar theorem of Host and Kra on ergodic actions of ${\Bbb Z}$, 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 $k \geq 0$ be an integer.  The concept of a polynomial $P: {\Bbb R} \to {\Bbb R}$ of one variable of degree $ (or $\leq k-1$) can be defined in one of two equivalent ways:

• (Global definition) $P: {\Bbb R} \to {\Bbb R}$ is a polynomial of degree $ iff it can be written in the form $P(x) = \sum_{0 \leq j < k} c_j x^j$ for some coefficients $c_j \in {\Bbb R}$.
• (Local definition) $P: {\Bbb R} \to {\Bbb R}$ is a polynomial of degree $ if it is k-times continuously differentiable and $\frac{d^k}{dx^k} P \equiv 0$.

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

$\displaystyle P(x) = \sum_{0 \leq j < k} \frac{P^{(j)}(0)}{j!} x^j$

we see that P is a polynomial in the global sense. We make the trivial remark that we have no difficulty dividing by $j!$ here, because the field ${\Bbb R}$ is of characteristic zero.

The above equivalence carries over to higher dimensions:

• (Global definition) $P: {\Bbb R}^n \to {\Bbb R}$ is a polynomial of degree $ iff it can be written in the form $P(x_1,\ldots,x_n) = \sum_{0 \leq j_1,\ldots,j_n; j_1+\ldots+j_n < k} c_{j_1,\ldots,j_n} x_1^{j_1} \ldots x_n^{j_n}$ for some coefficients $c_{j_1,\ldots,j_n} \in {\Bbb R}$.
• (Local definition) $P: {\Bbb R}^n \to {\Bbb R}$ is a polynomial of degree $ if it is k-times continuously differentiable and $(h_1 \cdot \nabla) \ldots (h_k \cdot \nabla) P \equiv 0$ for all $h_1,\ldots,h_k \in {\Bbb R}^n$.

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 $F^n$ 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 ${\Bbb R}/{\Bbb Z}$), the global definition needs to be corrected somewhat by adding some new monomials to the classical ones $x_1^{j_1} \ldots x_n^{j_n}$.  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 $p \geq k$, and gives a partial result in the low characteristic case $p < k$.

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).

Ben Green and I have just uploaded our joint paper, “The distribution of polynomials over finite fields, with applications to the Gowers norms“, to the arXiv, and submitted to Contributions to Discrete Mathematics. This paper, which we first announced at the recent FOCS meeting, and then gave an update on two weeks ago on this blog, is now in final form. It is being made available simultaneously with a closely related paper of Lovett, Meshulam, and Samorodnitsky.

In the previous post on this topic, I focused on the negative results in the paper, and in particular the fact that the inverse conjecture for the Gowers norm fails for certain degrees in low characteristic. Today, I’d like to focus instead on the positive results, which assert that for polynomials in many variables over finite fields whose degree is less than the characteristic of the field, one has a satisfactory theory for the distribution of these polynomials. Very roughly speaking, the main technical results are:

• A regularity lemma: Any polynomial can be expressed as a combination of a bounded number of other polynomials which are regular, in the sense that no non-trivial linear combination of these polynomials can be expressed efficiently in terms of lower degree polynomials.
• A counting lemma: A regular collection of polynomials behaves as if the polynomials were selected randomly. In particular, the polynomials are jointly equidistributed.

Recently, I had tentatively announced a forthcoming result with Ben Green establishing the “Gowers inverse conjecture” (or more accurately, the “inverse conjecture for the Gowers uniformity norm”) for vector spaces ${\Bbb F}_p^n$ over a finite field ${\Bbb F}_p$, in the special case when p=2 and when the function $f: {\Bbb F}_p^n \to {\Bbb C}$ for which the inverse conjecture is to be applied is assumed to be a polynomial phase of bounded degree (thus $f= e^{2\pi i P/|{\Bbb F}|}$, where $P: {\Bbb F}_p^n \to {\Bbb F}_p$ is a polynomial of some degree $d=O(1)$). See my FOCS article for some further discussion of this conjecture, which has applications to both polynomiality testing and to various structural decompositions involving the Gowers norm.

This conjecture can be informally stated as follows. By iterating the obvious fact that the derivative of a polynomial of degree at most d is a polynomial of degree at most d-1, we see that a function $P: {\Bbb F}_p^n \to {\Bbb F}_p$ is a polynomial of degree at most d if and only if

$\sum_{\omega_1,\ldots,\omega_{d+1} \in \{0,1\}} (-1)^{\omega_1+\ldots+\omega_{d+1}} P(x +\omega_1 h_1 + \ldots + \omega_{d+1} h_{d+1}) = 0$

for all $x,h_1,\ldots,h_{d+1} \in {\Bbb F}_p^n$. From this one can deduce that a function $f: {\Bbb F}_p^n \to {\Bbb C}$ bounded in magnitude by 1 is a polynomial phase of degree at most d if and only if the Gowers norm

$\|f\|_{U^{d+1}({\Bbb F}_p^n)} := \bigl( {\Bbb E}_{x,h_1,\ldots,h_{d+1} \in {\Bbb F}_p^n} \prod_{\omega_1,\ldots,\omega_{d+1} \in \{0,1\}}$

${\mathcal C}^{\omega_1+\ldots+\omega_{d+1}} f(x + \omega_1 h_1 + \ldots + \omega_{d+1} h_{d+1}) \bigr)^{1/2^{d+1}}$

is equal to its maximal value of 1. The inverse conjecture for the Gowers norm, in its usual formulation, says that, more generally, if a function $f: {\Bbb F}_p^n \to {\Bbb C}$ bounded in magnitude by 1 has large Gowers norm (e.g. $\|f\|_{U^{d+1}} \geq \varepsilon$) then f has some non-trivial correlation with some polynomial phase g (e.g. $\langle f, g \rangle > c(\varepsilon)$ for some $c(\varepsilon) > 0$). Informally, this conjecture asserts that if a function has biased $(d+1)^{th}$ derivatives, then one should be able to “integrate” this bias and conclude that the function is biased relative to a polynomial of degree d. The conjecture has already been proven for $d \leq 2$. There are analogues of this conjecture for cyclic groups which are of relevance to Szemerédi’s theorem and to counting linear patterns in primes, but I will not discuss those here.

At the time of the announcement, our paper had not quite been fully written up. This turned out to be a little unfortunate, because soon afterwards we discovered that our arguments at one point had to go through a version of Newton’s interpolation formula, which involves a factor of d! in the denominator and so is only valid when the characteristic p of the field exceeds the degree. So our arguments in fact are only valid in the range $p > d$, and in particular are rather trivial in the important case $p=2$; my previous announcement should thus be amended accordingly.

Earlier this month, in the previous incarnation of this page, I posed a question which I thought was unsolved, and obtained the answer (in fact, it was solved 25 years ago) within a week. Now that this new version of the page has better feedback capability, I am now tempted to try again, since I have a large number of such questions which I would like to publicise. (Actually, I even have a secret web page full of these somewhere near my home page, though it will take a non-trivial amount of effort to find it!)

Perhaps my favourite open question is the problem on the maximal size of a cap set – a subset of ${\Bbb F}^n_3$ (${\Bbb F}_3$ being the finite field of three elements) which contains no lines, or equivalently no non-trivial arithmetic progressions of length three. As an upper bound, one can easily modify the proof of Roth’s theorem to show that cap sets must have size $O(3^n/n)$ (see e.g. this paper of Meshulam). This of course is better than the trivial bound of $3^n$ once n is large. In the converse direction, the trivial example $\{0,1\}^n$ shows that cap sets can be as large as $2^n$; the current world record is $(2.2174\ldots)^n$, held by Edel. The gap between these two bounds is rather enormous; I would be very interested in either an improvement of the upper bound to $o(3^n/n)$, or an improvement of the lower bound to $(3-o(1))^n$. (I believe both improvements are true, though a good friend of mine disagrees about the improvement to the lower bound.)