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In the previous lectures, we have focused mostly on the equidistribution or linear patterns on a subset of the integers , and in particular on intervals . The integers are of course a very important domain to study in additive combinatorics; but there are also other fundamental model examples of domains to study. One of these is that of a vector space over a finite field of prime order. Such domains are of interest in computer science (particularly when ) and also in number theory; but they also serve as an important simplified “dyadic model” for the integers. See this survey article of Green for further discussion of this point.
The additive combinatorics of the integers , and of vector spaces over finite fields, are analogous, but not quite identical. For instance, the analogue of an arithmetic progression in is a subspace of . In many cases, the finite field theory is a little bit simpler than the integer theory; for instance, subspaces are closed under addition, whereas arithmetic progressions are only “almost” closed under addition in various senses. (For instance, is closed under addition approximately half of the time.) However, there are some ways in which the integers are better behaved. For instance, because the integers can be generated by a single generator, a homomorphism from to some other group can be described by a single group element : . However, to specify a homomorphism from a vector space to one would need to specify one group element for each dimension of . Thus we see that there is a tradeoff when passing from (or ) to a vector space model; one gains a bounded torsion property, at the expense of conceding the bounded generation property. (Of course, if one wants to deal with arbitrarily large domains, one has to concede one or the other; the only additive groups that have both bounded torsion and boundedly many generators, are bounded.)
The starting point for this course (Notes 1) was the study of equidistribution of polynomials from the integers to the unit circle. We now turn to the parallel theory of equidistribution of polynomials from vector spaces over finite fields to the unit circle. Actually, for simplicity we will mostly focus on the classical case, when the polynomials in fact take values in the roots of unity (where is the characteristic of the field ). As it turns out, the non-classical case is also of importance (particularly in low characteristic), but the theory is more difficult; see these notes for some further discussion.
Our study of random matrices, to date, has focused on somewhat general ensembles, such as iid random matrices or Wigner random matrices, in which the distribution of the individual entries of the matrices was essentially arbitrary (as long as certain moments, such as the mean and variance, were normalised). In these notes, we now focus on two much more special, and much more symmetric, ensembles:
- The Gaussian Unitary Ensemble (GUE), which is an ensemble of random Hermitian matrices in which the upper-triangular entries are iid with distribution , and the diagonal entries are iid with distribution , and independent of the upper-triangular ones; and
- The Gaussian random matrix ensemble, which is an ensemble of random (non-Hermitian) matrices whose entries are iid with distribution .
The symmetric nature of these ensembles will allow us to compute the spectral distribution by exact algebraic means, revealing a surprising connection with orthogonal polynomials and with determinantal processes. This will, for instance, recover the semi-circular law for GUE, but will also reveal fine spacing information, such as the distribution of the gap between adjacent eigenvalues, which is largely out of reach of tools such as the Stieltjes transform method and the moment method (although the moment method, with some effort, is able to control the extreme edges of the spectrum).
Similarly, we will see for the first time the circular law for eigenvalues of non-Hermitian matrices.
There are a number of other highly symmetric ensembles which can also be treated by the same methods, most notably the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). However, for simplicity we shall focus just on the above two ensembles. For a systematic treatment of these ensembles, see the text by Deift.
One of the most basic theorems in linear algebra is that every finite-dimensional vector space has a finite basis. Let us give a statement of this theorem in the case when the underlying field is the rationals:
Theorem 1 (Finite generation implies finite basis, infinitary version) Let be a vector space over the rationals , and let be a finite collection of vectors in . Then there exists a collection of vectors in , with , such that
- ( generates ) Every can be expressed as a rational linear combination of the .
- ( independent) There is no non-trivial linear relation , among the (where non-trivial means that the are not all zero).
In fact, one can take to be a subset of the .
Proof: We perform the following “rank reduction argument”. Start with initialised to (so initially we have ). Clearly generates . If the are linearly independent then we are done. Otherwise, there is a non-trivial linear relation between them; after shuffling things around, we see that one of the , say , is a rational linear combination of the . In such a case, becomes redundant, and we may delete it (reducing the rank by one). We repeat this procedure; it can only run for at most steps and so terminates with obeying both of the desired properties.
In additive combinatorics, one often wants to use results like this in finitary settings, such as that of a cyclic group where is a large prime. Now, technically speaking, is not a vector space over , because one only multiply an element of by a rational number if the denominator of that rational does not divide . But for very large, “behaves” like a vector space over , at least if one restricts attention to the rationals of “bounded height” – where the numerator and denominator of the rationals are bounded. Thus we shall refer to elements of as “vectors” over , even though strictly speaking this is not quite the case.
On the other hand, saying that one element of is a rational linear combination of another set of elements is not a very interesting statement: any non-zero element of already generates the entire space! However, if one again restricts attention to rational linear combinations of bounded height, then things become interesting again. For instance, the vector can generate elements such as or using rational linear combinations of bounded height, but will not be able to generate such elements of as without using rational numbers of unbounded height.
For similar reasons, the notion of linear independence over the rationals doesn’t initially look very interesting over : any two non-zero elements of are of course rationally dependent. But again, if one restricts attention to rational numbers of bounded height, then independence begins to emerge: for instance, and are independent in this sense.
Thus, it becomes natural to ask whether there is a “quantitative” analogue of Theorem 1, with non-trivial content in the case of “vector spaces over the bounded height rationals” such as , which asserts that given any bounded collection of elements, one can find another set which is linearly independent “over the rationals up to some height”, such that the can be generated by the “over the rationals up to some height”. Of course to make this rigorous, one needs to quantify the two heights here, the one giving the independence, and the one giving the generation. In order to be useful for applications, it turns out that one often needs the former height to be much larger than the latter; exponentially larger, for instance, is not an uncommon request. Fortunately, one can accomplish this, at the cost of making the height somewhat large:
Theorem 2 (Finite generation implies finite basis, finitary version) Let be an integer, and let be a function. Let be an abelian group which admits a well-defined division operation by any natural number of size at most for some constant depending only on ; for instance one can take for a prime larger than . Let be a finite collection of “vectors” in . Then there exists a collection of vectors in , with , as well an integer , such that
- (Complexity bound) for some depending only on .
- ( generates ) Every can be expressed as a rational linear combination of the of height at most (i.e. the numerator and denominator of the coefficients are at most ).
- ( independent) There is no non-trivial linear relation among the in which the are rational numbers of height at most .
In fact, one can take to be a subset of the .
Proof: We perform the same “rank reduction argument” as before, but translated to the finitary setting. Start with initialised to (so initially we have ), and initialise . Clearly generates at this height. If the are linearly independent up to rationals of height then we are done. Otherwise, there is a non-trivial linear relation between them; after shuffling things around, we see that one of the , say , is a rational linear combination of the , whose height is bounded by some function depending on and . In such a case, becomes redundant, and we may delete it (reducing the rank by one), but note that in order for the remaining to generate we need to raise the height upper bound for the rationals involved from to some quantity depending on . We then replace by and continue the process. We repeat this procedure; it can only run for at most steps and so terminates with and obeying all of the desired properties. (Note that the bound on is quite poor, being essentially an -fold iteration of ! Thus, for instance, if is exponential, then the bound on is tower-exponential in nature.)
(A variant of this type of approximate basis lemma was used in my paper with Van Vu on the singularity probability of random Bernoulli matrices.)
Looking at the statements and proofs of these two theorems it is clear that the two results are in some sense the “same” result, except that the latter has been made sufficiently quantitative that it is meaningful in such finitary settings as . In this note I will show how this equivalence can be made formal using the language of non-standard analysis. This is not a particularly deep (or new) observation, but it is perhaps the simplest example I know of that illustrates how nonstandard analysis can be used to transfer a quantifier-heavy finitary statement, such as Theorem 2, into a quantifier-light infinitary statement, such as Theorem 1, thus lessening the need to perform “epsilon management” duties, such as keeping track of unspecified growth functions such as . This type of transference is discussed at length in this previous blog post of mine.
In this particular case, the amount of effort needed to set up the nonstandard machinery in order to reduce Theorem 2 from Theorem 1 is too great for this transference to be particularly worthwhile, especially given that Theorem 2 has such a short proof. However, when performing a particularly intricate argument in additive combinatorics, in which one is performing a number of “rank reduction arguments”, “energy increment arguments”, “regularity lemmas”, “structure theorems”, and so forth, the purely finitary approach can become bogged down with all the epsilon management one needs to do to organise all the parameters that are flying around. The nonstandard approach can efficiently hide a large number of these parameters from view, and it can then become worthwhile to invest in the nonstandard framework in order to clean up the rest of a lengthy argument. Furthermore, an advantage of moving up to the infinitary setting is that one can then deploy all the firepower of an existing well-developed infinitary theory of mathematics (in this particular case, this would be the theory of linear algebra) out of the box, whereas in the finitary setting one would have to painstakingly finitise each aspect of such a theory that one wished to use (imagine for instance trying to finitise the rank-nullity theorem for rationals of bounded height).
The nonstandard approach is very closely related to use of compactness arguments, or of the technique of taking ultralimits and ultraproducts; indeed we will use an ultrafilter in order to create the nonstandard model in the first place.
I will also discuss a two variants of both Theorem 1 and Theorem 2 which have actually shown up in my research. The first is that of the regularity lemma for polynomials over finite fields, which came up when studying the equidistribution of such polynomials (in this paper with Ben Green). The second comes up when is dealing not with a single finite collection of vectors, but rather with a family of such vectors, where ranges over a large set; this gives rise to what we call the sunflower lemma, and came up in this recent paper of myself, Ben Green, and Tamar Ziegler.
This post is mostly concerned with nonstandard translations of the “rank reduction argument”. Nonstandard translations of the “energy increment argument” and “density increment argument” were briefly discussed in this recent post; I may return to this topic in more detail in a future post.
Ben Green, Tamar Ziegler and I have just uploaded to the arXiv our paper “An inverse theorem for the Gowers norm“. This paper establishes the next case of the inverse conjecture for the Gowers norm for the integers (after the 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 .
To state the inverse conjecture properly requires a certain amount of notation. Given a function and a shift , define the multiplicative derivative
and then define the Gowers norm of a function to (essentially) be the quantity
where we extend f by zero outside of . (Actually, we use a slightly different normalisation to ensure that the function 1 has a norm of 1, but never mind this for now.)
Informally, the Gowers norm measures the amount of bias present in the multiplicative derivatives of . In particular, if for some polynomial , then the derivative of 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 , 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 holding a good fraction of the time) that it turns out that third derivative is trivial fairly often, and the Gowers norm is comparable to 1. This bracket polynomial phase can be modeled as a nilsequence , where is a polynomial orbit on a nilmanifold , which in this case has step 2. (The function F is only piecewise smooth, due to the discontinuity in the floor function , 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 is bounded but has large norm. Then there is an s-step nilsequence 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 of large norm must correlate with a “bracket cubic phase”, which is the product of a bounded number of phases from the following list
for various real numbers .
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.
A (concrete) Boolean algebra is a pair , where X is a set, and is a collection of subsets of X which contain the empty set , and which is closed under unions , intersections , and complements . The subset relation also gives a relation on . Because the is concretely represented as subsets of a space X, these relations automatically obey various axioms, in particular, for any , we have:
- is a partial ordering on , and A and B have join and meet .
- We have the distributive laws and .
- is the minimal element of the partial ordering , and is the maximal element.
- and .
We can then define an abstract Boolean algebra to be an abstract set with the specified objects, operations, and relations that obey the axioms 1-4. [Of course, some of these operations are redundant; for instance, intersection can be defined in terms of complement and union by de Morgan's laws. In the literature, different authors select different initial operations and axioms when defining an abstract Boolean algebra, but they are all easily seen to be equivalent to each other. To emphasise the abstract nature of these algebras, the symbols are often replaced with other symbols such as .]
Clearly, every concrete Boolean algebra is an abstract Boolean algebra. In the converse direction, we have Stone’s representation theorem (see below), which asserts (among other things) that every abstract Boolean algebra is isomorphic to a concrete one (and even constructs this concrete representation of the abstract Boolean algebra canonically). So, up to (abstract) isomorphism, there is really no difference between a concrete Boolean algebra and an abstract one.
Now let us turn from Boolean algebras to -algebras.
A concrete -algebra (also known as a measurable space) is a pair , where X is a set, and is a collection of subsets of X which contains and are closed under countable unions, countable intersections, and complements; thus every concrete -algebra is a concrete Boolean algebra, but not conversely. As before, concrete -algebras come equipped with the structures which obey axioms 1-4, but they also come with the operations of countable union and countable intersection , which obey an additional axiom:
As with Boolean algebras, one can now define an abstract -algebra to be a set with the indicated objects, operations, and relations, which obeys axioms 1-5. Again, every concrete -algebra is an abstract one; but is it still true that every abstract -algebra is representable as a concrete one?
The answer turns out to be no, but the obstruction can be described precisely (namely, one needs to quotient out an ideal of “null sets” from the concrete -algebra), and there is a satisfactory representation theorem, namely the Loomis-Sikorski representation theorem (see below). As a corollary of this representation theorem, one can also represent abstract measure spaces (also known as measure algebras) by concrete measure spaces, , after quotienting out by null sets.
In the rest of this post, I will state and prove these representation theorems. They are not actually used directly in the rest of the course (and they will also require some results that we haven’t proven yet, most notably Tychonoff’s theorem), and so these notes are optional reading; but these theorems do help explain why it is “safe” to focus attention primarily on concrete -algebras and measure spaces when doing measure theory, since the abstract analogues of these mathematical concepts are largely equivalent to their concrete counterparts. (The situation is quite different for non-commutative measure theories, such as quantum probability, in which there is basically no good representation theorem available to equate the abstract with the classically concrete, but I will not discuss these theories 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 at Rutgers University, giving the Lewis Memorial Lectures for this year, which are also concurrently part of a workshop in random matrices. I gave four lectures, three of which were on random matrices, and one of which was on the Szemerédi regularity lemma.
The titles, abstracts, and slides of these talks are as follows.
- Szemerédi’s lemma revisited. In this general-audience talk, I discuss the Szemerédi regularity lemma (which, roughly speaking, shows that an arbitrary large dense graph can always be viewed as the disjoint union of a bounded number of pseudorandom components), and how it has recently been reinterpreted in a more analytical (and infinitary) language using the theory of graph limits or of exchangeable measures. I also discuss arithmetic analogues of this lemma, including one which (implicitly) underlies my result with Ben Green that the primes contain arbitrarily long arithmetic progressions.
- Singularity and determinant of random matrices. Here, I present recent progress in understanding the question of how likely a random matrix (e.g. one whose entries are all +1 or -1 with equal probability) is to be invertible, as well as the related question of how large the determinant should be. The case of continuous matrix ensembles (such as the Gaussian ensemble) is well understood, but the discrete case contains some combinatorial difficulties and took longer to understand properly. In particular I present the results of Kahn-Komlós-Szemerédi and later authors showing that discrete random matrices are invertible with exponentially high probability, and also give some results for the distribution of the determinant.
- The least singular value of random matrices. A more quantitative version of the question “when is a matrix invertible?” is “what is the least singular value of that matrix”? I present here the recent results of Litvak-Pajor-Rudelson-Tomczak-Jaegermann, Rudelson, myself and Vu, and Rudelson-Vershynin on addressing this question in the discrete case. A central role is played by the inverse Littlewood-Offord theorems of additive combinatorics, which give reasonably sharp necessary conditions for a discrete random walk to concentrate in a small ball.
- The circular law. One interesting application of the above theory is to extend the circular law for the spectrum of random matrices from the continuous case to the discrete case. Previous arguments of Girko and Bai for the continuous case can be transplanted to the discrete case, but the key new ingredient needed is a least singular value bound for shifted matrices in order to avoid the spectrum being overwhelmed by pseudospectrum. It turns out that the results of the preceding lecture are almost precisely what are needed to accomplish this.
[Update, Mar 31: first lecture slides corrected. Thanks to Yoshiyasu Ishigami for pointing out a slight inaccuracy in the text.]
In one of my recent posts, I used the Jordan normal form for a matrix in order to justify a couple of arguments. As a student, I learned the derivation of this form twice: firstly (as an undergraduate) by using the minimal polynomial, and secondly (as a graduate) by using the structure theorem for finitely generated modules over a principal ideal domain. I found though that the former proof was too concrete and the latter proof too abstract, and so I never really got a good intuition on how the theorem really worked. So I went back and tried to synthesise a proof that I was happy with, by taking the best bits of both arguments that I knew. I ended up with something which wasn’t too different from the standard proofs (relying primarily on the (extended) Euclidean algorithm and the fundamental theorem of algebra), but seems to get at the heart of the matter fairly quickly, so I thought I’d put it up on this blog anyway.
Before we begin, though, let us recall what the Jordan normal form theorem is. For this post, I’ll take the perspective of abstract linear transformations rather than of concrete matrices. Let be a linear transformation on a finite dimensional complex vector space V, with no preferred coordinate system. We are interested in asking what possible “kinds” of linear transformations V can support (more technically, we want to classify the conjugacy classes of , the ring of linear endomorphisms of V to itself). Here are some simple examples of linear transformations.
- The right shift. Here, is a standard vector space, and the right shift is defined as , thus all elements are shifted right by one position. (For instance, the 1-dimensional right shift is just the zero operator.)
- The right shift plus a constant. Here we consider an operator , where is a right shift, I is the identity on V, and is a complex number.
- Direct sums. Given two linear transformations and , we can form their direct sum by the formula .
Our objective is then to prove the
Jordan normal form theorem. Every linear transformation on a finite dimensional complex vector space V is similar to a direct sum of transformations, each of which is a right shift plus a constant.
(Of course, the same theorem also holds with left shifts instead of right shifts.)
In my discussion of the Oppenheim conjecture in my recent post on Ratner’s theorems, I mentioned in passing the simple but crucial fact that the (orthochronous) special orthogonal group of an indefinite quadratic form on can be generated by unipotent elements. This is not a difficult fact to prove, as one can simply diagonalise Q and then explicitly write down some unipotent elements (the magic words here are “null rotations“). But this is a purely algebraic approach; I thought it would also be instructive to show the geometric (or dynamic) reason for why unipotent elements appear in the orthogonal group of indefinite quadratic forms in three dimensions. (I’ll give away the punch line right away: it’s because the parabola is a conic section.) This is not a particularly deep or significant observation, and will not be surprising to the experts, but I would like to record it anyway, as it allows me to review some useful bits and pieces of elementary linear algebra.
This problem lies in the highly interconnected interface between algebraic combinatorics (esp. the combinatorics of Young tableaux and related objects, including honeycombs and puzzles), algebraic geometry (particularly classical and quantum intersection theory and geometric invariant theory), linear algebra (additive and multiplicative, real and tropical), and the representation theory (classical, quantum, crystal, etc.) of classical groups. (Another open problem in this subject is to find a succinct and descriptive name for the field.) I myself haven’t actively worked in this area for several years, but I still find it a fascinating and beautiful subject. (With respect to the dichotomy between structure and randomness, this subject lies deep within the “structure” end of the spectrum.)
As mentioned above, the problems in this area can be approached from a variety of quite diverse perspectives, but here I will focus on the linear algebra perspective, which is perhaps the most accessible. About nine years ago, Allen Knutson and I introduced a combinatorial gadget, called a honeycomb, which among other things controlled the relationship between the eigenvalues of two arbitrary Hermitian matrices A, B, and the eigenvalues of their sum A+B; this was not the first such gadget that achieved this purpose, but it was a particularly convenient one for studying this problem, in particular it was used to resolve two conjectures in the subject, the saturation conjecture and the Horn conjecture. (These conjectures have since been proven by a variety of other methods.) There is a natural multiplicative version of these problems, which now relates the eigenvalues of two arbitrary unitary matrices U, V and the eigenvalues of their product UV; this led to the “quantum saturation” and “quantum Horn” conjectures, which were proven a couple years ago. However, the quantum analogue of a “honeycomb” remains a mystery; this is the main topic of the current post.