I am writing a short survey on connections between additive combinatorics and computer science for SIGACT News and I have been wondering about the “history” of the connections. (I will be writing as little as possible about history in the SIGACT article, because I don’t have the time to research it carefully, but if readers help me out, I would write about it in a longer article that I plan to write sometime next year.)

Now, of course, there is always that Russian paper from the 1950s . . .

Then there is the Coppersmith-Winograd matrix multiplication algorithm, which uses Behrend’s construction of large sets of integers with no length-3 arithmetic progressions. (They don’t need the full strength of Behrend’s construction, and the weaker construction of Salem and Spencer suffices.) As far as I know, this was an isolated application.

My understanding (which is quite possibly mistaken) is that there are three main threads to the current interactions: one concerning the Szemeredi Regularity Lemma and the “triangle removal lemma” of Ruzsa and Szemeredi; one concerning “sum-product theorems” and the Kakeya problem; and one concerning the Gowers uniformity norms. The earliest connection is the first one, and, depending how one is counting, started in 1992 or 2001, and is due, respectively, to Alon et al. or just to Alon.

The Regularity Lemma, the Triangle Removal Lemma, and Graph Property Testing

The Szemeredi Regularity Lemma is a main technical tool in Szemeredi’s 1969 proof of Szemeredi’s theorem for progressions of length 4, and, eventually, his 1976 proof of the full 1975 proof of Szemeredi’s theorem. Rusza and Szemeredi, in a 1978 paper called “Triple systems with no six points carrying three triangles” showed that one could derive Roth’s Theorem (the length-3 case of Szemeredi’s theorem) as an simple consequence of a “Triangle Removal Lemma” (the derivation is simple but rather clever) which is itself a simple consequence of the Regularity Lemma. I have seen the Rusza-Szemeredi paper been referenced as either written in 1976 or 1978 – I never had a chance to see the paper itself. (More here about the Triangle Removal Lemma and Roth’s theorem.) Behrend had proved in the 1940s a negative result about the parameters in Roth’s theorem (before, indeed, Roth’s theorem was proved), which then, because of the Rusza-Szemeredi work, translates into a negative result about the parameters in the Triangle Removal Lemma: the lemma says that if we need to remove \Omega (n^2) edges from an n-vertex graph in order to remove all triangles, then there must be \Omega(n^3) triangles in total. The negative result is that the constants hidden in the \Omega() notations cannot be polynomially related.

There were various paper in the 1990s on applications of the Regularity Lemma, and of variants of it, in computer science, starting, I think, with a 1992 paper by Alon, Duke, Lefmann, Rödl, and Yuster. Notably, Frieze and Kannan proved in 1996 that the Regularity Lemma could be used to provide approximation schemes for optimization problems on dense graphs. Motivated by this application, Frieze and Kannan then formulated and proved the Weak Regularity Lemma, a version of the result that gives a weaker form of partition, but with much better quantitative bounds.

Coming to the main connection, after Goldreich, Goldwasser and Ron introduced the notion of property testing in 1996, various people realized that the GGR notion of testing graph properties in the adjacency matrix model captured earlier work in extremal combinatorics. A 1999 paper by Alon, Fischer, Krivelevich, and Szegedy shows how to use the Szemeredi Regularity Lemma to design property testing algorithms, and I think it is the first paper of the kind.

Two years later, Alon studied the problem of testing the property of having a fixed graph H as a subgraph, and showed that the problem is always testable, but that the complexity of the tester obeys a sharp dichotomy depending on whether the subgraph H is bipartite or not. Alon uses Behrend’s construction in his lower bound for the case of non-bipartite H, and, indeed, when H is a triangle Alon’s proof reduces to the Ruzsa-Szemeredi proof mentioned above.

The Rusza-Szemeredi argument has a natural analog that proves the full Szemeredi’s theorem, provided that one can prove a “clique removal lemma for hypergraphs,” which in turn one would want to derive from a properly formulated “Hypergraph Regularity Lemma.” This approach ended up being extremely difficult to follow, and such a proof was devised only a few years ago, independently by Nagle, Rödl, Schacht, and Skokan; by Gowers; and by Tao. This in turn has various applications to testing properties of dense hypergraphs, which have been studied by additive combinatorialists.

Kakeya Sets and Sum-Product Theorems

A Besicovitch set, or Kakeya set is a set that, for every direction, contains a unit-length segment parallel to that direction. The Kakeya problem is, roughly speaking, how “large” such a set must be. It is known that, for every n, there are Kakeya sets in {\mathbb R}^n of Lebesgue measure 0; if, however, one looks at more refined measures of “size,” the Kakeya conjecture is that the Hausdorff and Minkowski dimension of any Kakeya set is n, the largest possible. This has various applications in analysis and connections to other problems about which I know almost nothing; the interested reader will enjoy reading a fascinating survey paper by Terence Tao.

A particularly important turn of events started in1998, when Bourgain saw a way to make progress on this problem using one of the staples of additive combinatorics: an understanding of the structures of subsets A of a group such that A+A := \{ a+b : a\in A, b\in A\} is small. Then people realized that such techniques would also work, usually in a cleaner way, to prove lower bounds to the size of a Kakeya set in {\mathbb F}^n_p, a problem which is rather interesting in its own. In this more combinatorial setup, the Finite Field Kakeya Conjecture, formulated by Wolff in 1999, was that a Kakeya set in {\mathbb F}^n_p must have size at least \Omega_n (q^n), where the constant in the \Omega() notation depends on n but not on q. Motivated by this conjecture, Wolff made the finite field sum-product conjecture that for every subset A \subseteq {\mathbb F}_p, either |A+A| or |A \cdot A| is at least \min \{ p, |A|^{1+\epsilon} \} for an absolute positive constant \epsilon>0. If A is a set of integers, Erdos and Szemeredi had shown that this is true, but their proof (and later improvements) uses geometric properties that are not true for finite geometries.

Meanwhile, one of Avi Wigderson’s favorite open problems was that of finding “seedless” randomness extractors for independent sources. (That’s quite the abrupt switch of topic, but bear with me.) There had been no progress on it since the 1985 paper by Chor and Goldreich which had introduced the problem. A while ago (the 1990s, I believe) Avi formulated the following question: suppose X,Y,Z are independent random variables ranging over {\mathbb F}_p, each of (min-)entropy at least k; is it true that X\cdot Y + Z \bmod p much have min-entropy at least (1+\epsilon) \cdot k, for some absolute constant \epsilon>0? This would allow to make progress on the extractor problem.

Bourgain, Katz and Tao proved in 2004 the Wolff finite field sum-product conjecture. (For sets A that were not too small and not too large; this restriction was removed by Konyagin.) This was in the ballpark of Avi’s problem (in fact, the finite field sum-product conjecture had been made by Avi as well), and a 2004 paper by Barak, Impagliazzo and Wigderson proves Avi’s conjecture and provides the first progress on seedless extractors for independent sources since 1985. Rapid progress followed, which was quite dramatic, including a 2006 paper by Barak, Rao, Shaltiel, and Wigderson making the first progress since 1981 in constructing Ramsey graphs. (As a side note, while it would have in almost any other conference, the BRSW paper did not win the STOC’06 best paper award because of the incredible competition: at the same conference, there was the Daskalakis-Goldberg-Papadimitriou PPAD-completeness of Nash equilibria, and Irit Dinur’s combinatorial proof of the PCP Theorem. Our community has trouble pacing itself when it comes to breakthroughs…) Techniques from arithmetic combinatorics have played a key role in this amazing work.

But the most surprising turn of events regarded the Kakeya problem itself: Zeev Dvir, a complexity theorist, found an absolutely stunning three-page proof of the Kakeya conjecture in finite fields. (Which is now, I guess, Dvir’s theorem.) Zeev’s proof, which uses simple properties of low-degree polynomials in finite fields (a staple of complexity theory) is very much a “computer science proof.”

Gowers Norms

This is the thread that I have been involved with, so it’s probably the one in which I’ll get the story mixed up the most.

At the end of the 1990s, Gowers found an analytic proof of Szemeredi’s theorem, which had much better quantitative parameters than the two previously known proofs: Szemeredi’s own, and Furstenberg’s ergodic theoretic proof. The main technical tool introduced by Gowers was a new norm, now called the Gowers uniformity norm, for functions whose domain is an abelian group. Expressions similar to the one that defined the Gowers norm had occurred before in ergodic theory, in work motivated by Furstenberg’s proof of Szemeredi’s theorem. (In recent years, there has been a lot of progress in understanding the connections between Gowers’s work and the related work in ergodic theory.)

Perhaps more surprisingly, a somewhat similar expression had appeared in a 1993 paper by Chung and Tetali on communication complexity in the number-on-the-forehead model. More closely related, in 2003 Alon, Kaufman, Krivelevich, Litsyn and Ron wrote a paper on testing low-degree multivariate polynomials over {\mathbb F}_2. The acceptance probability of their tester is (up to some normalization) precisely the Gowers norm of the function being tested, and their result is that the tester has high acceptance probability (and so the Gowers norm is close to 1) if and only the function is close to a low degree polynomial.

Meanwhile, Green and Tao had been applying the Gowers uniformity norm in their spectacular proof that the primes contain arbitrarily long arithmetic progressions, had started a research program aimed at proving bounds on the number of solutions in primes of various types of systems of linear equations (arithmetic progressions are a particular case of patterns expressible via liner equations), and had also turned their attention to achieving further quantitative improvements in Szemeredi’s theorem, by improving various parts of Gowers’s argument.

Both in their work on patterns in primes and their work on quantitative bounds in Szemeredi’s theorem, Green and Tao found that a required step was to prove that if a function has a noticeably large Gowers norm, then it correlated with a “low degree polynomial type” object. In the case of functions whose domain is {\mathbb F}_p^n (where we think of p small and fixed and n being large) then the conjecture is simply that when the Gowers norm is noticeable, then the function has noticeable correlation with a low degree polynomial. They called this question the “Gowers norm inverse conjecture.” More precisely, for every k one can define a “k-th Gowers norm,” and the inverse conjecture is that noticeable k-th Gowers norm implies correlation with degree-(k-1) polynomials. This is trivial for k=1 and simple for k=2. In March 2005, Green and Tao proved the k=3 case when the domain is {\mathbb F}_p^n, with odd p. (And, in fact, for all abelian groups of odd order.)

When the Green-Tao paper on primes in arithmetic progressions came out in 2004, I downloaded it with the same spirit in which I donwloaded Wiles’s paper when it came out. I wanted to hold a piece of mathematical history in my hands, and begin to read the introduction. I assumed that by the second sentence I wouldn’t know what they were talking about, and by the third sentence I wouldn’t know which words were supposed to be nouns and which words were supposed to be verbs. Instead I made it through the end of the introduction, and I was hooked, so I started going back to some of related earlier papers to gain a bit of background, which made me realize I had to go back to read even more background papers and so on. I had recently gotten tenure, so this was something that I could afford to do. Also I was impressed by the work of Avi and others using sum-product theorems (described above), and there was a sense that more connections could be discovered between additive combinatorics and computer science, and the equivalence between Gowers norm and the low-degree test of Alon and others was one of the things I noticed. I should mention that people without tenure were also studying this material and noticing these connections. Ryan O’Donnell, for example, wrote an expository note in August 2005 about additive combinatorics and computational complexity which mentioned the equivalence of Gowers norm and low-degree test, and all the other connections mentioned above.

Anyways, with a stunning synchronicity, just as I had downloaded the Green-Tao paper on the Gowers norm inverse conjecture, in April 2005 Alex Samorodnitsky sent me a preprint showing his result that if the quadratic low-degree test of Alon et al. has acceptance probability bounded away from \frac 12, then the function is correlated with a degree-2 polynomial. This was the k=3 case of the inverse Gowers conjecture for functions whose domain is {\mathbb F}^n_2, a case that was missing from the Green-Tao paper. Amazingly, Alex’s work was entirely independent, and it was a tour de force: he proved the {\mathbb F}^n_2 case of the “Freiman-Rusza theorem” (a sort of “linearity test for matrices” about which Emanuele Viola has written an excellent expository note) and he came up with the idea of showing the existence of the correlated degree-2 polynomial via an “integration” approach. Alex also, in a different language, states the Gowers norm inverse conjecture in {\mathbb F}^n_2 for all k.

(About the remark on “integration:” Usually, low degree tests are analyzed in the following way: if the tested function f makes the test accept with sufficiently large probability, then one constructs a function g from f via some kind of “local decoding” procedure, then one proves that the function g is a low-degree polynomial, and that f and g are correlated. In Alex’s work, and also in the Green-Tao proof, the idea is to construct a degree-2 polynomial g such that the “derivatives” of g are correlated with the derivates of f, and then argue that g itself is correlated with f. This is a sort of an “integral” because we are constructing a function starting from the knowledge of its derivatives. Of course these are discrete analogs of derivatives, we only know an approximation of the derivatives, and the construction process looks nothing like integration in real or complex analysis.)

Alex was wondering whether his test for quadratic polynomials could be turned into a PCP construction. (A few years earlier, we had worked on a joint paper on PCP.) My reply was: (1) congratulations for having proved the Gowers norm inverse conjecture for k=3 in {\mathbb F}^n_2 and (2) for applications to PCP, instead of relating noticeable Gowers norm to correlation to polynomials, one should relate it to the influence of variables. Thus we started a collaboration to prove (2), which resulted in our joint work on Gowers norm and PCP constructions.

After our work, there have been a few more occurrences of Gowers norm in complexity theory. Viola and Wigderson use the Gowers norm to prove “XOR lemmas” and “direct product theorems” in communication complexity and for low-degree polynomials. In communication complexity, they use the results of Chung and Tetali mentioned above. In studying correlation with low degree polynomials (see the article by Viola in the January issue of SIGACT News) they use the low-degree test of Alon and others.

Bogdanov and Viola construct a pseudorandom generator for low-degree polynomials which is conditional to the Gowers inverse conjecture; in order to fool degree-d polynomials one needs the inverse conjecture for the d-th Gowers norm. (Alex’s result makes the result unconditional for quadratic and cubic polynomials.) Later, work by Lovett and Viola makes the result unconditional and avoids any explicit mention of the Gowers norm.

Instead, Lovett and Viola simply make a careful use of induction and Cauchy–Schwarz. Indeed, my paper with Alex also just uses induction and Cauchy-Schwarz, in connection with some simple properties of the Gowers norm that themselves have proofs that come down to induction and Cauchy–Schwarz. (It would be interesting to see a way to abstract the use of such techniques, and find other places in computer science where they can be useful.)

This story has two more twists. First, Lovett, Meshulam and Samorodnitsky, and independently Green and Tao, show that the inverse conjecture for the k-th Gowers norm in {\mathbb F}_2^n is false for k\geq 4. (Recall that it’s trivial for k=1, easy for k=2, and it was proved by Alex for k=3.)

Then, most recently, Bergelson, Tao, and Ziegler have proved the Gowers norm inverse conjecture for every k, in finite fields of sufficiently large characteristic. This is done in a series of two papers: one by Bergelson, Tao and Ziegler dealing with an infinitary analog, and another by Tao and Ziegler establishing a “correspondence principle” that transfers the result from the infinitary case to the finite case. They have also tentatively announced a proof for all characteristics, but with a different definition of “polynomial” in the case of small characteristic; this difference avoids the counterexamples mentioned above.

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