The Inverse Conjecture for the Gowers Norms

This semester, the MSRI is having a special semester devoted to additive combinatorics and ergodic theory.

Additive combinatorics is the branch of extremal combinatorics where the objects of interest are sets of integers and, more generally, subsets of abelian groups (rather than graphs or set systems) and where the properties of interest are formulated in terms of linear equations (rather than in terms of cuts, partitions, intersections, and so on). Lately, it has been quite fascinating for a convergence of methods from “classical” combinatorics, from analysis and from ergodic theory. I have often written about it here because the combinatorial and analytical techniques of additive combinatorics have been useful in a number of different computer science applications (related to probabilistically checkable proof constructions, property testing, and pseudorandomness), and computer scientists are becoming more and more interested in the subject, contributing their own perspective to it.

In all this, the exact role of ergodic theory (and the possible applications of ergodic-theoretic techniques in complexity theory) has been a bit of mystery to me, and perhaps to other computer scientists too. Very nicely, the MSRI special program started this week with a series of tutorials to introduce the connections between ergodic theory and additive combinatorics.

All talks are (or will soon be) online, and the talks by Terry Tao are especially recommended, because he explains, using very simple examples, how one goes about converting concrete, finitary and quantitative statements into equivalent abstract infinitary statements, which are in turn amenable to ergodic-theoretic techniques.

Today, Tamar Ziegler discussed the very recent proof, by Vitaly Bergelson, Terry Tao, and herself, of the inverse conjecture for Gowers norms in finite fields, a proof that uses ergodic-theoretic techniques.

But, you may object, didn’t we know that the inverse conjecture for Gowers norms is false? Well, the counterexample of Lovett, Meshulam, and Samorodnitsky (independently discovered by Green and Tao) refutes the following conjecture: “Suppose $f: {\mathbb F}_p^n \rightarrow {\mathbb C}$ is a bounded function such that $|| f||_{U^k} \geq \delta$; then there is a $\epsilon = \epsilon(\delta,p,k)$ independent of $n$ and an $n$-variate degree-(k-1) polynomial $q$ over ${\mathbb F}_p$ such that $f(x)$ and $e^{-2\pi i q(x)/p}$ have correlation at least $\epsilon$.”

This is refuted in the $p=2,k=4$ case by taking $f(x_1,\ldots,x_n) = (-1)^{S_4(x_1,\ldots,x_n)}$, where $S_4$ is the symmetric polynomial of degree 4. While $f$ has $o(1)$ (indeed, exponentially small) correlation with all functions of the form $(-1)^{q(x_1,\ldots,x_n)}$, where $q$ is a degree-3 polynomial, the norm $|| f||_{U^4}$ is a constant.

The statement proved by Bergelson, Tao, and Ziegler is “Suppose $f: {\mathbb F}_p^n \rightarrow {\mathbb C}$ is a bounded function such that $|| f||_{U^k} \geq \delta$; then there is a $\epsilon = \epsilon(\delta,p,k)$ independent of $n$ and a bounded $n$-variate degree-(k-1) ‘polynomial function’ $Q: {\mathbb F}_p^n \rightarrow {\mathbb C}$ such that $f(x)$ and $Q(x)$ have correlation at least $\epsilon$.”

What is, then, a ‘polynomial function’ of degree $d$? It is a function bounded by 1 in absolute value, and such that for every $d+1$ directions $h_1,\ldots, h_{d+1}$, if one takes $d+1$ Gowers derivatives in such directions one always gets the constant-1 function. In other words, $Q$ is a ‘polynomial function’ of degree $k-1$ if $|Q(x)| \leq 1$ for every $x$, and one has $|| Q||_{U^k}=1$. Interestingly, these functions are a proper superclass of the functions of the form $e^{\pi i q(x)/p}$ with $q$ being a polynomial over ${\mathbb F}_p$.

In the concrete $p=2$ case, one may construct such a function by letting $q$ be a polynomial in the ring, say, ${\mathbb Z}_8$, and then having $Q(x) = \omega^{q(x)}$, where $\omega$ is a primitive eight-root of unity. Indeed, this is the type of degree-3 polynomial function that is correlated with $(-1)^{S_4(x)}$.

[Apologies for not defining all the technical terms and the context; the reader can find some background in this post and following the links there.]

What is, then, ergodic theory, and what does it have to do with finitary combinatorial problems? I am certainly the wrong person to ask, but I shall try to explain the little that I have understood in the next post(s).