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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 is a bounded function such that ; then there is a independent of and an -variate degree-(k-1) polynomial over such that and have correlation at least .”
This is refuted in the case by taking , where is the symmetric polynomial of degree 4. While has (indeed, exponentially small) correlation with all functions of the form , where is a degree-3 polynomial, the norm is a constant.
The statement proved by Bergelson, Tao, and Ziegler is “Suppose is a bounded function such that ; then there is a independent of and a bounded -variate degree-(k-1) ‘polynomial function’ such that and have correlation at least .”
What is, then, a ‘polynomial function’ of degree ? It is a function bounded by 1 in absolute value, and such that for every directions , if one takes Gowers derivatives in such directions one always gets the constant-1 function. In other words, is a ‘polynomial function’ of degree if for every , and one has . Interestingly, these functions are a proper superclass of the functions of the form with being a polynomial over .
In the concrete case, one may construct such a function by letting be a polynomial in the ring, say, , and then having , where is a primitive eight-root of unity. Indeed, this is the type of degree-3 polynomial function that is correlated with .
[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).
In 1990, Linial and Nisan conjectured that every AC0 function is “fooled” by polylog-wise independence. This means that if is computed by a circuit of size and depth , and is a distribution over such that every bits are mutually independent and uniformly distributed (such a distribution is called -wise independent), then
where is the uniform distribution over , provided that is around . (Maybe even around .)
Nearly 20 years later, not much is known about this conjecture, except for a breakthrough result by Louay Bazzi from last year. Bazzi proves that (1) holds if is computed by a size- depth-2 circuit (that is, a CNF or a DNF) and is a -wise independent distribution. This settles the depth-2 case of the Linial-Nisan conjecture.
A result of Linial, Mansour and Nisan plays an important role in the proof. The result is that if is computed by a DNF of size , then there is a function that can be written as a degree- polynomial with and that is “close” to , in the sense that
Now, if is a degree- polynomial, and is a -wise independent distribution, we have
and we also have from (2). If we also had , then we would be done.
Unfortunately, we have no way to relate and , because the fact that and are close does not imply that they have similar expectations under the distribution , which could be concentrated on the few inputs on which and are very different.
Although the Linial-Mansour-Nisan result does not immediately give us the desired result, it is intuitive that it is useful to be able to approximate a DNF by low-degree polynomials, and to know that low-degree polynomials are perfectly “fooled” by -wise independent distributions.
Bazzi observes that the following stronger polynomial approximation property is sufficient to prove that a distribution fools a given function.
Suppose that is a function, and that we are able to find two degree- polynomials such that for every we have
where the expectation in (4) is taken under the uniform distribution. Then it is easy to see that if is -wise independent, then
It is also possible to use linear programming duality that argue that if is fooled by -wise independent distributions, then degree- polynomials as above must exist.
Bazzi then shows that, given a size- DNF , it is possible to find degree- polynomials as above with .
Finding such polynomials, in turn, is reduced rather cleverly to the task of finding a single polynomial such that is small, as in Linial-Mansour-Nisan and, in addition, has the property that whenever then we always must have .
The construction of this “one sided” low-degree approximation of is the bulk of Bazzi’s paper. A couple of days ago, Razborov has posted an exceedingly neat construction, that he explains in a page and a half! (Theorem 7 in the link above.)
It is rather appropriate that this has happened during the Banff Workshop on Analytic Tools in Computational Complexity, and tomorrow Scott Aaronson will give an impromptu talk on Razborov’s proof, hot from the presses.