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Ben Green, Tamar Ziegler and I have just uploaded to the arXiv our paper “An inverse theorem for the Gowers $U^4$ norm“.  This paper establishes the next case of the inverse conjecture for the Gowers norm for the integers (after the $U^3$ 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 $[N] = \{1,\ldots,N\}$.

To state the inverse conjecture properly requires a certain amount of notation.  Given a function $f: {\Bbb Z} \to {\Bbb C}$ and a shift $h \in {\Bbb Z}$, define the multiplicative derivative

$\Delta_h f(x) := f(x+h) \overline{f(x)}$

and then define the Gowers $U^{s+1}[N]$ norm of a function $f: [N] \to {\Bbb C}$ to (essentially) be the quantity

$\| f\|_{U^{s+1}[N]} := ({\Bbb E}_{h_1,\ldots,h_{s+1} \in [-N,N]} {\Bbb E}_{x \in [N]} |\Delta_{h_1} \ldots \Delta_{h_{s+1}} f(x)|)^{1/2^{s+1}},$

where we extend f by zero outside of $[N]$.  (Actually, we use a slightly different normalisation to ensure that the function 1 has a $U^{s+1}$ norm of 1, but never mind this for now.)

Informally, the Gowers norm $\|f\|_{U^{s+1}[N]}$ measures the amount of bias present in the $(s+1)^{st}$ multiplicative derivatives of $f$.  In particular, if $f = e(P) := e^{2\pi i P}$ for some polynomial $P: {\Bbb Z} \to {\Bbb C}$, then the $(s+1)^{th}$ derivative of $f$ 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 $f(n) := e( \lfloor \sqrt{2} n \rfloor \sqrt{3} n )$, 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 $\lfloor x+y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor$ holding a good fraction of the time) that it turns out that third derivative is trivial fairly often, and the Gowers norm $\|f\|_{U^3[N]}$ is comparable to 1.  This bracket polynomial phase can be modeled as a nilsequence $n \mapsto F( g(n) \Gamma )$, where $n \mapsto g(n) \Gamma$ is a polynomial orbit on a nilmanifold $G/\Gamma$, which in this case has step 2.  (The function F is only piecewise smooth, due to the discontinuity in the floor function $\lfloor \rfloor$, 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 $f: [N] \to {\Bbb C}$ is bounded but has large $U^{s+1}[N]$ norm.  Then there is an s-step nilsequence $n \mapsto F( g(n) \Gamma )$ 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 $f$ of large $U^4$ norm must correlate with a “bracket cubic phase”, which is the product of a bounded number of phases from the following list

$e( \alpha n^3 + \beta n^2 + \gamma n), e( \lfloor \alpha n \rfloor \beta n^2 ), e( \lfloor \alpha n \rfloor \lfloor \beta n \rfloor \gamma n ), e( \lfloor \alpha n \rfloor \beta n )$ (*)

for various real numbers $\alpha,\beta,\gamma$.

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.

For a number of reasons, including the start of the summer break for me and my coauthors, a number of papers that we have been working on are being released this week.  For instance, Ben Green and I have just uploaded to the arXiv our paper “An equivalence between inverse sumset theorems and inverse conjectures for the $U^3$ norm“, submitted to Math. Proc. Camb. Phil. Soc..  The main result of this short paper (which was briefly announced in this earlier post) is a connection between two types of inverse theorems in additive combinatorics, namely the inverse sumset theorems of Freiman type, and inverse theorems for the Gowers uniformity norm, and more specifically, for the $U^3$ norm

$\|f\|_{U^3(G)}^8 := {\Bbb E}_{x,a,b,c \in G} f(x) \overline{f(x+a)} \overline{f(x+b)} \overline{f(x+c)} f(x+a+b) f(x+a+c) f(x+b+c) \overline{f(x+a+b+c)}$

on finite additive group G, where $f: G \to {\Bbb C}$ is a complex-valued function.

As usual, the connection is easiest to state in a finite field model such as $G = {\Bbb F}_2^n$.  In this case, we have the following inverse sumset theorem of Ruzsa:

Theorem 1. If $A \subset {\Bbb F}_2^n$ is such that $|A+A| \leq K|A|$, then A can be covered by a translate of a subspace of ${\Bbb F}_2^n$ of cardinality at most $K^2 2^{K^4} |A|$.

The constant $K^2 2^{K^4}$ has been improved for large $K$ in a sequence of papers, from $K 2^{\lfloor K^3 \rfloor-1}$ by Ruzsa, $K^2 2^{K^2-2}$ by Green-Ruzsa, $2^{O(K^{3/2} \log(1+K)}$ by Sanders, $2^{2K+O(\sqrt{K} \log K})$ by Green and myself, and finally $2^{2K+O(\log K)}$ by Konyagin (private communication) which is sharp except for the precise value of the O() implied constant (as can be seen by considering the example when A consists of about 2K independent elements).  However, it is conjectured that the polynomial loss can be removed entirely if one modifies the conclusion slightly:

Conjecture 1. (Polynomial Freiman-Ruzsa conjecture for ${\Bbb F}_2^n$.) If $A \subset {\Bbb F}_2^n$ is such that $|A+A| \leq K|A|$, then A can be covered by $O(K^{O(1)})$ translates of subspaces of ${\Bbb F}_2^n$ of cardinality at most |A|.

This conjecture was verified for downsets by Green and myself, but is open in general.   This conjecture has a number of equivalent formulations; see this paper of Green for more discussion.  In this previous post we show that a stronger version of this conjecture fails.

Meanwhile, for the Gowers norm, we have the following inverse theorem, due to Samorodnitsky:

Theorem 2. Let $f: {\Bbb F}_2^n \to [-1,1]$ be a function whose $U^3$ norm is at least 1/K.  Then there exists a quadratic polynomial $Q: {\Bbb F}_2^n \to {\Bbb F}_2$ such that $|{\Bbb E}_{x \in {\Bbb F}_2^n} f(x) (-1)^{Q(x)}| \geq \exp( - O(K)^{O(1)} )$.

Note that the quadratic phases $(-1)^{Q(x)}$ are the only functions taking values in [-1,1] whose $U^3$ norm attains its maximal value of 1.

It is conjectured that the exponentially weak correlation here can be strengthened to a polynomial one:

Conjecture 2. (Polynomial inverse conjecture for the $U^3({\Bbb F}_2^n)$ norm). Let $f: {\Bbb F}_2^n \to [-1,1]$ be a function whose $U^3$ norm is at least 1/K.  Then there exists a quadratic polynomial $Q: {\Bbb F}_2^n \to {\Bbb F}_2$ such that $|{\Bbb E}_{x \in {\Bbb F}_2^n} f(x) (-1)^{Q(x)}| \geq K^{-O(1)}$.

The first main result of this paper is

Theorem 3. Conjecture 1 and Conjecture 2 are equivalent.

This result was also independently observed by Shachar Lovett (private communication).  We also establish an analogous result for the cyclic group ${\Bbb Z}/N{\Bbb Z}$, in which the notion of polynomial is replaced by that of a subexponential $\exp(K^{o(1)})$, and in which the notion of a quadratic polynomial is replaced by a 2-step nilsequence; the precise statement is a bit technical and will not be given here.  We also observe a partial partial analogue of the correpsondence between inverse sumset theorems and Gowers norms in the higher order case, in particular observing that $U^4$ inverse theorems imply a certain rigidity result for “Freiman-quadratic polynomials” (a quadratic version of Conjecture 3 below).

Below the fold, we sketch the proof of Theorem 3.

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.)

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).