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A few days ago, I received the sad news that Yahya Ould Hamidoune had recently died. Hamidoune worked in additive combinatorics, and had recently solved a question on noncommutative Freiman-Kneser theorems posed by myself on this blog last year. Namely, Hamidoune showed
Theorem 1 (Noncommutative Freiman-Kneser theorem for small doubling) Let , and let be a finite non-empty subset of a multiplicative group such that for some finite set of cardinality at least , where is the product set of and . Then there exists a finite subgroup of with cardinality , such that is covered by at most right-cosets of , where depend only on .
One can of course specialise here to the case , and view this theorem as a classification of those sets of doubling constant at most .
In fact Hamidoune’s argument, which is completely elementary, gives the very nice explicit constants and , which are essentially optimal except for factors of (as can be seen by considering an arithmetic progression in an additive group). This result was also independently established (in the case) by Tom Sanders (unpublished) by a more Fourier-analytic method, in particular drawing on Sanders’ deep results on the Wiener algebra on arbitrary non-commutative groups .
Theorem 1 is not, strictly speaking, contained in Hamidoune’s paper, but can be extracted from his arguments, which share some similarity with the recent simple proof of the Ruzsa-Plünnecke inequality by Petridis (as discussed by Tim Gowers here), and this is what I would like to do below the fold. I also include (with permission) Sanders’ unpublished argument, which proceeds instead by Fourier-analytic methods. Read the rest of this entry »
In Notes 3, we saw that the number of additive patterns in a given set was (in principle, at least) controlled by the Gowers uniformity norms of functions associated to that set.
Such norms can be defined on any finite additive group (and also on some other types of domains, though we will not discuss this point here). In particular, they can be defined on the finite-dimensional vector spaces over a finite field .
In this case, the Gowers norms are closely tied to the space of polynomials of degree at most . Indeed, as noted in Exercise 20 of Notes 4, a function of norm has norm equal to if and only if for some ; thus polynomials solve the “ inverse problem” for the trivial inequality . They are also a crucial component of the solution to the “ inverse problem” and “ inverse problem”. For the former, we will soon show:
Thus, for the Gowers norm to be almost completely saturated, one must be very close to a polynomial. The converse assertion is easily established:
Exercise 1 (Converse to inverse theorem for ) If and for some , then , where is a constant depending only on .
In the world, one no longer expects to be close to a polynomial. Instead, one expects to correlate with a polynomial. Indeed, one has
Lemma 2 (Converse to the inverse theorem for ) If and are such that , where , then .
Proof: From the definition of the norm (equation (18) from Notes 3), the monotonicity of the Gowers norms (Exercise 19 of Notes 3), and the polynomial phase modulation invariance of the Gowers norms (Exercise 21 of Notes 3), one has
and the claim follows.
In the high characteristic case at least, this can be reversed:
This result is sometimes referred to as the inverse conjecture for the Gowers norm (in high, but bounded, characteristic). For small , the claim is easy:
Exercise 2 Verify the cases of this theorem. (Hint: to verify the case, use the Fourier-analytic identities and , where is the space of all homomorphisms from to , and are the Fourier coefficients of .)
This conjecture for larger values of are more difficult to establish. The case of the theorem was established by Ben Green and myself in the high characteristic case ; the low characteristic case was independently and simultaneously established by Samorodnitsky. The cases in the high characteristic case was established in two stages, firstly using a modification of the Furstenberg correspondence principle, due to Ziegler and myself. to convert the problem to an ergodic theory counterpart, and then using a modification of the methods of Host-Kra and Ziegler to solve that counterpart, as done in this paper of Bergelson, Ziegler, and myself.
The situation with the low characteristic case in general is still unclear. In the high characteristic case, we saw from Notes 4 that one could replace the space of non-classical polynomials in the above conjecture with the essentially equivalent space of classical polynomials . However, as we shall see below, this turns out not to be the case in certain low characteristic cases (a fact first observed by Lovett, Meshulam, and Samorodnitsky, and independently by Ben Green and myself), for instance if and ; this is ultimately due to the existence in those cases of non-classical polynomials which exhibit no significant correlation with classical polynomials of equal or lesser degree. This distinction between classical and non-classical polynomials appears to be a rather non-trivial obstruction to understanding the low characteristic setting; it may be necessary to obtain a more complete theory of non-classical polynomials in order to fully settle this issue.
The inverse conjecture has a number of consequences. For instance, it can be used to establish the analogue of Szemerédi’s theorem in this setting:
Exercise 3 Use Theorem 4 to establish the following generalisation: with the notation as above, if and is sufficiently large depending on , then contains an affine -dimensional subspace.
We will prove this theorem in two different ways, one using a density increment method, and the other using an energy increment method. We discuss some other applications below the fold.
A handy inequality in additive combinatorics is the Plünnecke-Ruzsa inequality:
The proof uses graph-theoretic techniques. Setting , we obtain a useful corollary: if has small doubling in the sense that , then we have for all , where is the sum of copies of .
In a recent paper, I adapted a number of sum set estimates to the entropy setting, in which finite sets such as in are replaced with discrete random variables taking values in , and (the logarithm of) cardinality of a set is replaced by Shannon entropy of a random variable . (Throughout this note I assume all entropies to be finite.) However, at the time, I was unable to find an entropy analogue of the Plünnecke-Ruzsa inequality, because I did not know how to adapt the graph theory argument to the entropy setting.
In fact Theorem 2 is a bit “better” than Theorem 1 in the sense that Theorem 1 needed to refine the original set to a subset , but no such refinement is needed in Theorem 2. One corollary of Theorem 2 is that if , then for all , where are independent copies of ; this improves slightly over the analogous combinatorial inequality. Indeed, the function is concave (this can be seen by using the version of Theorem 2 (or (2) below) to show that the quantity is decreasing in ).
Theorem 2 is actually a quick consequence of the submodularity inequality
in information theory, which is valid whenever are discrete random variables such that and each determine (i.e. is a function of , and also a function of ), and and jointly determine (i.e is a function of and ). To apply this, let be independent discrete random variables taking values in . Observe that the pairs and each determine , and jointly determine . Applying (1) we conclude that
and Theorem 2 follows by telescoping series.
The proof of Theorem 2 seems to be genuinely different from the graph-theoretic proof of Theorem 1. It would be interesting to see if the above argument can be somehow adapted to give a stronger version of Theorem 1. Note also that both Theorem 1 and Theorem 2 have extensions to more general combinations of than ; see this paper and this paper respectively.
Below the fold is a version of my talk “Recent progress on the Kakeya conjecture” that I gave at the Fefferman conference.
Additive combinatorics is largely focused on the additive properties of finite subsets A of an additive group . This group can be finite or infinite, but there is a very convenient trick, the Ruzsa projection trick, which allows one to reduce the latter case to the former. For instance, consider the set inside the integers . The integers of course form an infinite group, but if we are only interested in sums of at most two elements of A at a time, we can embed A ininside the finite cyclic group without losing any combinatorial information. More precisely, there is a Freiman isomorphism of order 2 between the set in and the set in . One can view the latter version of as a model for the former version of . More generally, it turns out that any finite set A in an additive group can be modeled in the above set by an equivalent set in a finite group, and in fact one can ensure that this ambient modeling group is not much larger than A itself if A has some additive structure; see this paper of Ruzsa (or Lemma 5.26 of my book with Van Vu) for a precise statement. This projection trick has a number of important uses in additive combinatorics, most notably in Ruzsa’s simplified proof of Freiman’s theorem.
Given the interest in non-commutative analogues of Freiman’s theorem, it is natural to ask whether one can similarly model finite sets A in multiplicative (and non-commutative) groups using finite models. Unfortunately (as I learned recently from Akshay Venkatesh, via Ben Green), this turns out to be impossible in general, due to an old example of Higman. More precisely, Higman shows:
Theorem 1. There exists an infinite group G generated by four distinct elements a,b,c,d that obey the relations
in fact, a and c generate the free group in G. On the other hand, if G’ is a finite group containing four elements a,b,c,d obeying (1), then a,b,c,d are all trivial.
As a consequence, the finite set in G has no model (in the sense of Freiman isomorphisms) in a finite group.
Theorem 1 is proven by a small amount of elementary group theory and number theory, and it was neat enough that I thought I would reproduce it here.
I’ve uploaded a new paper to the arXiv entitled “The sum-product phenomenon in arbitrary rings“, and submitted to Contributions to Discrete Mathematics. The sum-product phenomenon asserts, very roughly speaking, that given a finite non-empty set A in a ring R, then either the sum set or the product set will be significantly larger than A, unless A is somehow very close to being a subring of R, or if A is highly degenerate (for instance, containing a lot of zero divisors). For instance, in the case of the integers , which has no non-trivial finite subrings, a long-standing conjecture of Erdös and Szemerédi asserts that for every finite non-empty and every . (The current best result on this problem is a very recent result of Solymosi, who shows that the conjecture holds for any greater than 2/3.) In recent years, many other special rings have been studied intensively, most notably finite fields and cyclic groups, but also the complex numbers, quaternions, and other division algebras, and continuous counterparts in which A is now (for instance) a collection of intervals on the real line. I will not try to summarise all the work on sum-product estimates and their applications (which range from number theory to graph theory to ergodic theory to computer science) here, but I discuss this in the introduction to my paper, which has over 50 references to this literature (and I am probably still missing out on a few).
I was recently asked the question as to what could be said about the sum-product phenomenon in an arbitrary ring R, which need not be commutative or contain a multiplicative identity. Once one makes some assumptions to avoid the degenerate case when A (or related sets, such as A-A) are full of zero-divisors, it turns out that there is in fact quite a bit one can say, using only elementary methods from additive combinatorics (in particular, the Plünnecke-Ruzsa sum set theory). Roughly speaking, the main results of the paper assert that in an arbitrary ring, a set A which is non-degenerate and has small sum set and product set must be mostly contained inside a subring of R of comparable size to A, or a dilate of such a subring, though in the absence of invertible elements one sometimes has to enlarge the ambient ring R slightly before one can find the subring. At the end of the paper I specialise these results to specific rings, such as division algebras or products of division algebras, cyclic groups, or finite-dimensional algebras over fields. Generally speaking, the methods here give very good results when the set of zero divisors is sparse and easily describable, but poor results otherwise. (In particular, the sum-product theory in cyclic groups, as worked out by Bourgain and coauthors, is only recovered for groups which are the product of a bounded number of primes; the case of cyclic groups whose order has many factors seems to require a more multi-scale analysis which I did not attempt to perform in this paper.)
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This week I am visiting the University of Washington in Seattle, giving the Milliman Lecture Series for 2007-2008. My chosen theme here is “Recent developments in arithmetic combinatorics“. In my first lecture, I will speak (once again) on how methods in additive combinatorics have allowed us to detect additive patterns in the prime numbers, in particular discussing my joint work with Ben Green. In the second lecture I will discuss how additive combinatorics has made it possible to study the invertibility and spectral behaviour of random discrete matrices, in particular discussing my joint work with Van Vu; and in the third lecture I will discuss how sum-product estimates have recently led to progress in the theory of expanders relating to Lie groups, as well as to sieving over orbits of such groups, in particular presenting work of Jean Bourgain and his coauthors.
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 over a finite field , in the special case when p=2 and when the function for which the inverse conjecture is to be applied is assumed to be a polynomial phase of bounded degree (thus , where is a polynomial of some degree ). 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 is a polynomial of degree at most d if and only if
for all . From this one can deduce that a function bounded in magnitude by 1 is a polynomial phase of degree at most d if and only if the Gowers norm
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 bounded in magnitude by 1 has large Gowers norm (e.g. ) then f has some non-trivial correlation with some polynomial phase g (e.g. for some ). Informally, this conjecture asserts that if a function has biased 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 . 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 , and in particular are rather trivial in the important case ; my previous announcement should thus be amended accordingly.