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Let be a finite-dimensional Lie algebra (over the reals). Given two sufficiently small elements of , define the *right Baker-Campbell-Hausdorff-Dynkin law*

where , is the adjoint map , and is the function , which is analytic for near . Similarly, define the *left Baker-Campbell-Hausdorff-Dynkin law*

where . One easily verifies that these expressions are well-defined (and depend smoothly on and ) when and are sufficiently small.

We have the famous Baker-Campbell-Hausdoff-Dynkin formula:

Theorem 1 (BCH formula)Let be a finite-dimensional Lie group over the reals with Lie algebra . Let be a local inverse of the exponential map , defined in a neighbourhood of the identity. Then for sufficiently small , one has

See for instance these notes of mine for a proof of this formula (it is for , but one easily obtains a similar proof for ).

In particular, one can give a neighbourhood of the identity in the structure of a local Lie group by defining the group operation as

for sufficiently small , and the inverse operation by (one easily verifies that for all small ).

It is tempting to reverse the BCH formula and conclude (the local form of) *Lie’s third theorem*, that every finite-dimensional Lie algebra is isomorphic to the Lie algebra of some local Lie group, by using (3) to define a smooth local group structure on a neighbourhood of the identity. (See this previous post for a definition of a local Lie group.) The main difficulty in doing so is in verifying that the definition (3) is well-defined (i.e. that is always equal to ) and locally associative. The well-definedness issue can be trivially disposed of by using just one of the expressions or as the definition of (though, as we shall see, it will be very convenient to use both of them simultaneously). However, the associativity is not obvious at all.

With the assistance of Ado’s theorem, which places inside the general linear Lie algebra for some , one can deduce both the well-definedness and associativity of (3) from the Baker-Campbell-Hausdorff formula for . However, Ado’s theorem is rather difficult to prove (see for instance this previous blog post for a proof), and it is natural to ask whether there is a way to establish these facts without Ado’s theorem.

After playing around with this for some time, I managed to extract a direct proof of well-definedness and local associativity of (3), giving a proof of Lie’s third theorem independent of Ado’s theorem. This is not a new result by any means, (indeed, the original proofs of Lie and Cartan of Lie’s third theorem did not use Ado’s theorem), but I found it an instructive exercise to work out the details, and so I am putting it up on this blog in case anyone else is interested (and also because I want to be able to find the argument again if I ever need it in the future).

Jordan’s theorem is a basic theorem in the theory of finite linear groups, and can be formulated as follows:

Theorem 1 (Jordan’s theorem)Let be a finite subgroup of the general linear group . Then there is an abelian subgroup of of index , where depends only on .

Informally, Jordan’s theorem asserts that finite linear groups over the complex numbers are almost abelian. The theorem can be extended to other fields of characteristic zero, and also to fields of positive characteristic so long as the characteristic does not divide the order of , but we will not consider these generalisations here. A proof of this theorem can be found for instance in these lecture notes of mine.

I recently learned (from this comment of Kevin Ventullo) that the finiteness hypothesis on the group in this theorem can be relaxed to the significantly weaker condition of periodicity. Recall that a group is periodic if all elements are of finite order. Jordan’s theorem with “finite” replaced by “periodic” is known as the Jordan-Schur theorem.

The Jordan-Schur theorem can be quickly deduced from Jordan’s theorem, and the following result of Schur:

Theorem 2 (Schur’s theorem)Every finitely generated periodic subgroup of a general linear group is finite. (Equivalently, every periodic linear group is locally finite.)

Remark 1The question of whetherallfinitely generated periodic subgroups (not necessarily linear in nature) were finite was known as the Burnside problem; the answer was shown to be negative by Golod and Shafarevich in 1964.

Let us see how Jordan’s theorem and Schur’s theorem combine via a compactness argument to form the Jordan-Schur theorem. Let be a periodic subgroup of . Then for every finite subset of , the group generated by is finite by Theorem 2. Applying Jordan’s theorem, contains an abelian subgroup of index at most .

In particular, given any finite number of finite subsets of , we can find abelian subgroups of respectively such that each has index at most in . We claim that we may furthermore impose the compatibility condition whenever . To see this, we set , locate an abelian subgroup of of index at most , and then set . As is covered by at most cosets of , we see that is covered by at most cosets of , and the claim follows.

Note that for each , the set of possible is finite, and so the product space of all configurations , as ranges over finite subsets of , is compact by Tychonoff’s theorem. Using the finite intersection property, we may thus locate a subgroup of of index at most for *all* finite subsets of , obeying the compatibility condition whenever . If we then set , where ranges over all finite subsets of , we then easily verify that is abelian and has index at most in , as required.

Below I record a proof of Schur’s theorem, which I extracted from this book of Wehrfritz. This was primarily an exercise for my own benefit, but perhaps it may be of interest to some other readers.

Let be a Lie group with Lie algebra . As is well known, the exponential map is a local homeomorphism near the identity. As such, the group law on can be locally pulled back to an operation defined on a neighbourhood of the identity in , defined as

where is the local inverse of the exponential map. One can view as the group law expressed in local exponential coordinates around the origin.

An asymptotic expansion for is provided by the Baker-Campbell-Hausdorff (BCH) formula

for all sufficiently small , where is the Lie bracket. More explicitly, one has the *Baker-Campbell-Hausdorff-Dynkin formula*

for all sufficiently small , where , is the adjoint representation , and is the function

which is real analytic near and can thus be applied to linear operators sufficiently close to the identity. One corollary of this is that the multiplication operation is real analytic in local coordinates, and so every smooth Lie group is in fact a real analytic Lie group.

It turns out that one does not need the full force of the smoothness hypothesis to obtain these conclusions. It is, for instance, a classical result that regularity of the group operations is already enough to obtain the Baker-Campbell-Hausdorff formula. Actually, it turns out that we can weaken this a bit, and show that even regularity (i.e. that the group operations are continuously differentiable, and the derivatives are locally Lipschitz) is enough to make the classical derivation of the Baker-Campbell-Hausdorff formula work. More precisely, we have

Theorem 1 ( Baker-Campbell-Hausdorff formula)Let be a finite-dimensional vector space, and suppose one has a continuous operation defined on a neighbourhood around the origin, which obeys the following three axioms:

- (Approximate additivity) For sufficiently close to the origin, one has
- (Associativity) For sufficiently close to the origin, .
- (Radial homogeneity) For sufficiently close to the origin, one has
for all . (In particular, for all sufficiently close to the origin.)

Then is real analytic (and in particular, smooth) near the origin. (In particular, gives a neighbourhood of the origin the structure of a local Lie group.)

Indeed, we will recover the Baker-Campbell-Hausdorff-Dynkin formula (after defining appropriately) in this setting; see below the fold.

The reason that we call this a Baker-Campbell-Hausdorff formula is that if the group operation has regularity, and has as an identity element, then Taylor expansion already gives (2), and in exponential coordinates (which, as it turns out, can be defined without much difficulty in the category) one automatically has (3).

We will record the proof of Theorem 1 below the fold; it largely follows the classical derivation of the BCH formula, but due to the low regularity one will rely on tools such as telescoping series and Riemann sums rather than on the fundamental theorem of calculus. As an application of this theorem, we can give an alternate derivation of one of the components of the solution to Hilbert’s fifth problem, namely the construction of a Lie group structure from a Gleason metric, which was covered in the previous post; we discuss this at the end of this article. With this approach, one can avoid any appeal to von Neumann’s theorem and Cartan’s theorem (discussed in this post), or the Kuranishi-Gleason extension theorem (discussed in this post).

Recall that a (complex) abstract Lie algebra is a complex vector space (either finite or infinite dimensional) equipped with a bilinear antisymmetric form that obeys the Jacobi identity

(One can of course define Lie algebras over other fields than the complex numbers , but in order to avoid some technical issues we shall work solely with the complex case in this post.)

An important special case of the abstract Lie algebras are the *concrete Lie algebras*, in which is a vector space of linear transformations on a vector space (which again can be either finite or infinite dimensional), and the bilinear form is given by the usual Lie bracket

It is easy to verify that every concrete Lie algebra is an abstract Lie algebra. In the converse direction, we have

Theorem 1Every abstract Lie algebra is isomorphic to a concrete Lie algebra.

To prove this theorem, we introduce the useful algebraic tool of the universal enveloping algebra of the abstract Lie algebra . This is the free (associative, complex) algebra generated by (viewed as a complex vector space), subject to the constraints

This algebra is described by the Poincaré-Birkhoff-Witt theorem, which asserts that given an ordered basis of as a vector space, that a basis of is given by “monomials” of the form

where is a natural number, the are an increasing sequence of indices in , and the are positive integers. Indeed, given two such monomials, one can express their product as a finite linear combination of further monomials of the form (3) after repeatedly applying (2) (which we rewrite as ) to reorder the terms in this product modulo lower order terms until one all monomials have their indices in the required increasing order. It is then a routine exercise in basic abstract algebra (using all the axioms of an abstract Lie algebra) to verify that this is multiplication rule on monomials does indeed define a complex associative algebra which has the universal properties required of the universal enveloping algebra.

The abstract Lie algebra acts on its universal enveloping algebra by left-multiplication: , thus giving a map from to . It is easy to verify that this map is a Lie algebra homomorphism (so this is indeed an action (or representation) of the Lie algebra), and this action is clearly faithful (i.e. the map from to is injective), since each element of maps the identity element of to a different element of , namely . Thus is isomorphic to its image in , proving Theorem 1.

In the converse direction, every representation of a Lie algebra “factors through” the universal enveloping algebra, in that it extends to an algebra homomorphism from to , which by abuse of notation we shall also call .

One drawback of Theorem 1 is that the space that the concrete Lie algebra acts on will almost always be infinite-dimensional, even when the original Lie algebra is finite-dimensional. However, there is a useful theorem of Ado that rectifies this:

Theorem 2 (Ado’s theorem)Every finite-dimensional abstract Lie algebra is isomorphic to a concrete Lie algebra over afinite-dimensionalvector space .

Among other things, this theorem can be used (in conjunction with the Baker-Campbell-Hausdorff formula) to show that every abstract (finite-dimensional) Lie group (or abstract local Lie group) is locally isomorphic to a linear group. (It is well-known, though, that abstract Lie groups are not necessarily *globally* isomorphic to a linear group, but we will not discuss these global obstructions here.)

Ado’s theorem is surprisingly tricky to prove in general, but some special cases are easy. For instance, one can try using the adjoint representation of on itself, defined by the action ; the Jacobi identity (1) ensures that this indeed a representation of . The kernel of this representation is the centre . This already gives Ado’s theorem in the case when is semisimple, in which case the center is trivial.

The adjoint representation does not suffice, by itself, to prove Ado’s theorem in the non-semisimple case. However, it does provide an important reduction in the proof, namely it reduces matters to showing that every finite-dimensional Lie algebra has a finite-dimensional representation which is faithful on the centre . Indeed, if one has such a representation, one can then take the direct sum of that representation with the adjoint representation to obtain a new finite-dimensional representation which is now faithful on all of , which then gives Ado’s theorem for .

It remains to find a finite-dimensional representation of which is faithful on the centre . In the case when is abelian, so that the centre is all of , this is again easy, because then acts faithfully on by the infinitesimal shear maps . In matrix form, this representation identifies each in this abelian Lie algebra with an “upper-triangular” matrix:

This construction gives a faithful finite-dimensional representation of the centre of any finite-dimensional Lie algebra. The standard proof of Ado’s theorem (which I believe dates back to work of Harish-Chandra) then proceeds by gradually “extending” this representation of the centre to larger and larger sub-algebras of , while preserving the finite-dimensionality of the representation and the faithfulness on , until one obtains a representation on the entire Lie algebra with the required properties. (For technical inductive reasons, one also needs to carry along an additional property of the representation, namely that it maps the nilradical to nilpotent elements, but we will discuss this technicality later.)

This procedure is a little tricky to execute in general, but becomes simpler in the nilpotent case, in which the lower central series becomes trivial for sufficiently large :

Theorem 3 (Ado’s theorem for nilpotent Lie algebras)Let be a finite-dimensional nilpotent Lie algebra. Then there exists a finite-dimensional faithful representation of . Furthermore, there exists a natural number such that , i.e. one has for all .

The second conclusion of Ado’s theorem here is useful for induction purposes. (By Engel’s theorem, this conclusion is also equivalent to the assertion that every element of is nilpotent, but we can prove Theorem 3 without explicitly invoking Engel’s theorem.)

Below the fold, I give a proof of Theorem 3, and then extend the argument to cover the full strength of Ado’s theorem. This is not a new argument – indeed, I am basing this particular presentation from the one in Fulton and Harris – but it was an instructive exercise for me to try to extract the proof of Ado’s theorem from the more general structural theory of Lie algebras (e.g. Engel’s theorem, Lie’s theorem, Levi decomposition, etc.) in which the result is usually placed. (However, the proof I know of still needs Engel’s theorem to establish the solvable case, and the Levi decomposition to then establish the general case.)

I’ve just uploaded to the arXiv my paper “Outliers in the spectrum of iid matrices with bounded rank perturbations“, submitted to Probability Theory and Related Fields. This paper is concerned with outliers to the *circular law* for iid random matrices. Recall that if is an matrix whose entries are iid complex random variables with mean zero and variance one, then the complex eigenvalues of the normalised matrix will almost surely be distributed according to the circular law distribution in the limit . (See these lecture notes for further discussion of this law.)

The circular law is also stable under bounded rank perturbations: if is a deterministic rank matrix of polynomial size (i.e. of operator norm ), then the circular law also holds for (this is proven in a paper of myself, Van Vu, and Manjunath Krisnhapur). In particular, the bulk of the eigenvalues (i.e. of the eigenvalues) will lie inside the unit disk .

However, this leaves open the possibility for one or more *outlier* eigenvalues that lie significantly outside the unit disk; the arguments in the paper cited above give some upper bound on the number of such eigenvalues (of the form for some absolute constant ) but does not exclude them entirely. And indeed, numerical data shows that such outliers can exist for certain bounded rank perturbations.

In this paper, some results are given as to when outliers exist, and how they are distributed. The easiest case is of course when there is no bounded rank perturbation: . In that case, an old result of Bai and Yin and of Geman shows that the spectral radius of is almost surely , thus all eigenvalues will be contained in a neighbourhood of the unit disk, and so there are no significant outliers. The proof is based on the moment method.

Now we consider a bounded rank perturbation which is nonzero, but which has a bounded operator norm: . In this case, it turns out that the matrix will have outliers if the deterministic component has outliers. More specifically (and under the technical hypothesis that the entries of have bounded fourth moment), if is an eigenvalue of with , then (for large enough), will almost surely have an eigenvalue at , and furthermore these will be the only outlier eigenvalues of .

Thus, for instance, adding a bounded nilpotent low rank matrix to will not create any outliers, because the nilpotent matrix only has eigenvalues at zero. On the other hand, adding a bounded Hermitian low rank matrix will create outliers as soon as this matrix has an operator norm greater than .

When I first thought about this problem (which was communicated to me by Larry Abbott), I believed that it was quite difficult, because I knew that the eigenvalues of non-Hermitian matrices were quite unstable with respect to general perturbations (as discussed in this previous blog post), and that there were no interlacing inequalities in this case to control bounded rank perturbations (as discussed in this post). However, as it turns out I had arrived at the wrong conclusion, especially in the exterior of the unit disk in which the resolvent is actually well controlled and so there is no pseudospectrum present to cause instability. This was pointed out to me by Alice Guionnet at an AIM workshop last week, after I had posed the above question during an open problems session. Furthermore, at the same workshop, Percy Deift emphasised the point that the basic determinantal identity

for matrices and matrices was a particularly useful identity in random matrix theory, as it converted problems about large () matrices into problems about small () matrices, which was particularly convenient in the regime when and was fixed. (Percy was speaking in the context of invariant ensembles, but the point is in fact more general than this.)

From this, it turned out to be a relatively simple manner to transform what appeared to be an intractable matrix problem into quite a well-behaved matrix problem for bounded . Specifically, suppose that had rank , so that one can factor for some (deterministic) matrix and matrix . To find an eigenvalue of , one has to solve the characteristic polynomial equation

This is an determinantal equation, which looks difficult to control analytically. But we can manipulate it using (1). If we make the assumption that is outside the spectrum of (which we can do as long as is well away from the unit disk, as the unperturbed matrix has no outliers), we can divide by to arrive at

Now we apply the crucial identity (1) to rearrange this as

The crucial point is that this is now an equation involving only a determinant, rather than an one, and is thus much easier to solve. The situation is particularly simple for rank one perturbations

in which case the eigenvalue equation is now just a scalar equation

that involves what is basically a single coefficient of the resolvent . (It is also an instructive exercise to derive this eigenvalue equation directly, rather than through (1).) There is by now a very well-developed theory for how to control such coefficients (particularly for in the exterior of the unit disk, in which case such basic tools as Neumann series work just fine); in particular, one has precise enough control on these coefficients to obtain the result on outliers mentioned above.

The same method can handle some other bounded rank perturbations. One basic example comes from looking at iid matrices with a non-zero mean and variance ; this can be modeled by where is the unit vector . Here, the bounded rank perturbation has a large operator norm (equal to ), so the previous result does not directly apply. Nevertheless, the self-adjoint nature of the perturbation has a stabilising effect, and I was able to show that there is still only one outlier, and that it is at the expected location of .

If one moves away from the case of self-adjoint perturbations, though, the situation changes. Let us now consider a matrix of the form , where is a randomised version of , e.g. , where the are iid Bernoulli signs; such models were proposed recently by Rajan and Abbott as a model for neural networks in which some nodes are excitatory (and give columns with positive mean) and some are inhibitory (leading to columns with negative mean). Despite the superficial similarity with the previous example, the outlier behaviour is now quite different. Instead of having one extremely large outlier (of size ) at an essentially deterministic location, we now have a number of eigenvalues of size , scattered according to a random process. Indeed, (in the case when the entries of were real and bounded) I was able to show that the outlier point process converged (in the sense of converging -point correlation functions) to the zeroes of a random Laurent series

where are iid real Gaussians. This is basically because the coefficients of the resolvent have a Neumann series whose coefficients enjoy a central limit theorem.

On the other hand, as already observed numerically (and rigorously, in the gaussian case) by Rajan and Abbott, if one projects such matrices to have row sum zero, then the outliers all disappear. This can be explained by another appeal to (1); this projection amounts to right-multiplying by the projection matrix to the zero-sum vectors. But by (1), the non-zero eigenvalues of the resulting matrix are the same as those for . Since annihilates , we thus see that in this case the bounded rank perturbation plays no role, and the question reduces to obtaining a circular law with no outliers for . As it turns out, this can be done by invoking the machinery of Van Vu and myself that we used to prove the circular law for various random matrix models.

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 .

(More succinctly: is a lattice which is distributive, bounded, and complemented.)

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:

5. Any countable family of elements of has supremum and infimum .

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.)

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