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I’ve just finished writing the first draft of my third book coming out of the 2010 blog posts, namely “Higher order Fourier analysis“, which was based primarily on my graduate course in the topic, though it also contains material from some additional posts related to linear and higher order Fourier analysis on the blog. It is available online here. As usual, comments and corrections are welcome. There is also a stub page for the book, which at present does not contain much more than the above link.
In this, the final lecture notes of this course, we discuss one of the motivating applications of the theory developed thus far, namely to count solutions to linear equations in primes (or in dense subsets of primes ). Unfortunately, the most famous linear equations in primes: the twin prime equation and the even Goldbach equation – remain out of reach of this technology (because the relevant affine linear forms involved are commensurate, and thus have infinite complexity with respect to the Gowers norms), but most other systems of equations, in particular that of arithmetic progressions for (or equivalently, for ) , as well as the odd Goldbach equation , are tractable.
To illustrate the main ideas, we will focus on the following result of Green:
Theorem 1 (Roth’s theorem in the primes) Let be a subset of primes whose upper density is positive. Then contains infinitely many arithmetic progressions of length three.
This should be compared with Roth’s theorem in the integers (Notes 2), which is the same statement but with the primes replaced by the integers (or natural numbers ). Indeed, Roth’s theorem for the primes is proven by transferring Roth’s theorem for the integers to the prime setting; the latter theorem is used as a “black box”. The key difficulty here in performing this transference is that the primes have zero density inside the integers; indeed, from the prime number theorem we have .
There are a number of generalisations of this transference technique. In a paper of Green and myself, we extended the above theorem to progressions of longer length (thus transferring Szemerédi’s theorem to the primes). In a series of papers (culminating in a paper to appear shortly) of Green, myself, and also Ziegler, related methods are also used to obtain an asymptotic for the number of solutions in the primes to any system of linear equations of bounded complexity. This latter result uses the full power of higher order Fourier analysis, in particular relying heavily on the inverse conjecture for the Gowers norms; in contrast, Roth’s theorem and Szemerédi’s theorem in the primes are “softer” results that do not need this conjecture.
To transfer results from the integers to the primes, there are three basic steps:
- A general transference principle, that transfers certain types of additive combinatorial results from dense subsets of the integers to dense subsets of a suitably “pseudorandom set” of integers (or more precisely, to the integers weighted by a suitably “pseudorandom measure”);
- An application of sieve theory to show that the primes (or more precisely, an affine modification of the primes) lie inside a suitably pseudorandom set of integers (or more precisely, have significant mass with respect to a suitably pseudorandom measure).
- If one is seeking asymptotics for patterns in the primes, and not simply lower bounds, one also needs to control correlations between the primes (or proxies for the primes, such as the Möbius function) with various objects that arise from higher order Fourier analysis, such as nilsequences.
The former step can be accomplished in a number of ways. For progressions of length three (and more generally, for controlling linear patterns of complexity at most one), transference can be accomplished by Fourier-analytic methods. For more complicated patterns, one can use techniques inspired by ergodic theory; more recently, simplified and more efficient methods based on duality (the Hahn-Banach theorem) have also been used. No number theory is used in this step. (In the case of transference to genuinely random sets, rather than pseudorandom sets, similar ideas appeared earlier in the graph theory setting, see this paper of Kohayakawa, Luczak, and Rodl.
The second step is accomplished by fairly standard sieve theory methods (e.g. the Selberg sieve, or the slight variants of this sieve used by Goldston and Yildirim). Remarkably, very little of the formidable apparatus of modern analytic number theory is needed for this step; for instance, the only fact about the Riemann zeta function that is truly needed is that it has a simple pole at , and no knowledge of L-functions is needed.
The third step does draw more significantly on analytic number theory techniques and results (most notably, the method of Vinogradov to compute oscillatory sums over the primes, and also the Siegel-Walfisz theorem that gives a good error term on the prime number theorem in arithemtic progressions). As these techniques are somewhat orthogonal to the main topic of this course, we shall only touch briefly on this aspect of the transference strategy.
In Notes 5, we saw that the Gowers uniformity norms on vector spaces in high characteristic were controlled by classical polynomial phases .
Now we study the analogous situation on cyclic groups . Here, there is an unexpected surprise: the polynomial phases (classical or otherwise) are no longer sufficient to control the Gowers norms once exceeds . To resolve this problem, one must enlarge the space of polynomials to a larger class. It turns out that there are at least three closely related options for this class: the local polynomials, the bracket polynomials, and the nilsequences. Each of the three classes has its own strengths and weaknesses, but in my opinion the nilsequences seem to be the most natural class, due to the rich algebraic and dynamical structure coming from the nilpotent Lie group undergirding such sequences. For reasons of space we shall focus primarily on the nilsequence viewpoint here.
Traditionally, nilsequences have been defined in terms of linear orbits on nilmanifolds ; however, in recent years it has been realised that it is convenient for technical reasons (particularly for the quantitative “single-scale” theory) to generalise this setup to that of polynomial orbits , and this is the perspective we will take here.
A polynomial phase on a finite abelian group is formed by starting with a polynomial to the unit circle, and then composing it with the exponential function . To create a nilsequence , we generalise this construction by starting with a polynomial into a nilmanifold , and then composing this with a Lipschitz function . (The Lipschitz regularity class is convenient for minor technical reasons, but one could also use other regularity classes here if desired.) These classes of sequences certainly include the polynomial phases, but are somewhat more general; for instance, they almost include bracket polynomial phases such as . (The “almost” here is because the relevant functions involved are only piecewise Lipschitz rather than Lipschitz, but this is primarily a technical issue and one should view bracket polynomial phases as “morally” being nilsequences.)
In these notes we set out the basic theory for these nilsequences, including their equidistribution theory (which generalises the equidistribution theory of polynomial flows on tori from Notes 1) and show that they are indeed obstructions to the Gowers norm being small. This leads to the inverse conjecture for the Gowers norms that shows that the Gowers norms on cyclic groups are indeed controlled by these sequences.
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.
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.
In the previous lecture notes, we used (linear) Fourier analysis to control the number of three-term arithmetic progressions in a given set . The power of the Fourier transform for this problem ultimately stemmed from the identity
for any cyclic group and any subset of that group (analogues of this identity also exist for other finite abelian groups, and to a lesser extent to non-abelian groups also, although that is not the focus of my current discussion). As it turns out, linear Fourier analysis is not able to discern higher order patterns, such as arithmetic progressions of length four; we give some demonstrations of this below the fold, taking advantage of the polynomial recurrence theory from Notes 1.
The main objective of this course is to introduce the (still nascent) theory of higher order Fourier analysis, which is capable of studying higher order patterns. The full theory is still rather complicated (at least, at our present level of understanding). However, one aspect of the theory is relatively simple, namely that we can largely reduce the study of arbitrary additive patterns to the study of a single type of additive pattern, namely the parallelopipeds
These patterns are particularly pleasant to handle, thanks to the large number of symmetries available on the discrete cube . For instance, whereas establishing the presence of arbitrarily long arithmetic progressions in dense sets is quite difficult (Szemerédi’s theorem), establishing arbitrarily high-dimensional parallelopipeds is much easier:
Exercise 1 Let be such that for some . If is sufficiently large depending on , show that there exists an integer such that . (Hint: obtain upper and lower bounds on the set .)
Exercise 2 (Hilbert cube lemma) Let be such that for some , and let be an integer. Show that if is sufficiently large depending on , then contains a parallelopiped of the form (2), with positive integers. (Hint: use the previous exercise and induction.) Conclude that if has positive upper density, then it contains infinitely many such parallelopipeds for each .
Exercise 3 Show that if is an integer, and is sufficiently large depending on , then for any parallelopiped (2) in the integers , there exists , not all zero, such that . (Hint: pigeonhole the in the residue classes modulo .) Use this to conclude that if is the set of all integers such that for all integers , then is a set of positive upper density (and also positive lower density) which does not contain any infinite parallelopipeds (thus one cannot take in the Hilbert cube lemma).
with a function on a finite abelian group , and is the complex conjugation operator; analogues of this norm also exist for group-like objects such as the progression , and also for measure-preserving systems (where they are known as the Gowers-Host-Kra uniformity seminorms, see this paper of Host-Kra for more discussion). In this set of notes we will focus on the basic properties of these norms; the deepest fact about them, known as the inverse conjecture for these norms, will be discussed in later notes.
We now give a basic application of Fourier analysis to the problem of counting additive patterns in sets, namely the following famous theorem of Roth:
is positive. Then contains infinitely many arithmetic progressions of length three, with and .
This is the first non-trivial case of Szemerédi’s theorem, which is the same assertion but with length three arithmetic progressions replaced by progressions of length for any .
As it turns out, one can prove Roth’s theorem by an application of linear Fourier analysis – by comparing the set (or more precisely, the indicator function of that set, or of pieces of that set) against linear characters for various frequencies . There are two extreme cases to consider (which are model examples of a more general dichotomy between structure and randomness). One is when is aligned up almost completely with one of these linear characters, for instance by being a Bohr set of the form
or more generally of the form
for some multi-dimensional frequency and some open set . In this case, arithmetic progressions can be located using the equidistribution theory of the previous set of notes. At the other extreme, one has Fourier-uniform or Fourier-pseudorandom sets, whose correlation with any linear character is negligible. In this case, arithmetic progressions can be produced in abundance via a Fourier-analytic calculation.
To handle the general case, one must somehow synthesise together the argument that deals with the structured case with the argument that deals with the random case. There are several known ways to do this, but they can be basically classified into two general methods, namely the density increment argument (or increment argument) and the energy increment argument (or increment argument).
The idea behind the density increment argument is to introduce a dichotomy: either the object being studied is pseudorandom (in which case one is done), or else one can use the theory of the structured objects to locate a sub-object of significantly higher “density” than the original object. As the density cannot exceed one, one should thus be done after a finite number of iterations of this dichotomy. This argument was introduced by Roth in his original proof of the above theorem.
The idea behind the energy increment argument is instead to decompose the original object into two pieces (and, sometimes, a small additional error term): a structured component that captures all the structured objects that have significant correlation with , and a pseudorandom component which has no significant correlation with any structured object. This decomposition usually proceeds by trying to maximise the “energy” (or norm) of the structured component, or dually by trying to minimise the energy of the residual between the original object and the structured object. This argument appears for instance in the proof of the Szemerédi regularity lemma (which, not coincidentally, can also be used to prove Roth’s theorem), and is also implicit in the ergodic theory approach to such problems (through the machinery of conditional expectation relative to a factor, which is a type of orthogonal projection, the existence of which is usually established via an energy increment argument). However, one can also deploy the energy increment argument in the Fourier analytic setting, to give an alternate Fourier-analytic proof of Roth’s theorem that differs in some ways from the density increment proof.
In these notes we give both two Fourier-analytic proofs of Roth’s theorem, one proceeding via the density increment argument, and the other by the energy increment argument. As it turns out, both of these arguments extend to establish Szemerédi’s theorem, and more generally in counting other types of patterns, but this is non-trivial (requiring some sort of inverse conjecture for the Gowers uniformity norms in both cases); we will discuss this further in later notes.
(Linear) Fourier analysis can be viewed as a tool to study an arbitrary function on (say) the integers , by looking at how such a function correlates with linear phases such as , where is the fundamental character, and is a frequency. These correlations control a number of expressions relating to , such as the expected behaviour of on arithmetic progressions of length three.
In this course we will be studying higher-order correlations, such as the correlation of with quadratic phases such as , as these will control the expected behaviour of on more complex patterns, such as arithmetic progressions of length four. In order to do this, we must first understand the behaviour of exponential sums such as
Such sums are closely related to the distribution of expressions such as in the unit circle , as varies from to . More generally, one is interested in the distribution of polynomials of one or more variables taking values in a torus ; for instance, one might be interested in the distribution of the quadruplet as both vary from to . Roughly speaking, once we understand these types of distributions, then the general machinery of quadratic Fourier analysis will then allow us to understand the distribution of the quadruplet for more general classes of functions ; this can lead for instance to an understanding of the distribution of arithmetic progressions of length in the primes, if is somehow related to the primes.
More generally, to find arithmetic progressions such as in a set , it would suffice to understand the equidistribution of the quadruplet in as and vary. This is the starting point for the fundamental connection between combinatorics (and more specifically, the task of finding patterns inside sets) and dynamics (and more specifically, the theory of equidistribution and recurrence in measure-preserving dynamical systems, which is a subfield of ergodic theory). This connection was explored in one of my previous classes; it will also be important in this course (particularly as a source of motivation), but the primary focus will be on finitary, and Fourier-based, methods.
The theory of equidistribution of polynomial orbits was developed in the linear case by Dirichlet and Kronecker, and in the polynomial case by Weyl. There are two regimes of interest; the (qualitative) asymptotic regime in which the scale parameter is sent to infinity, and the (quantitative) single-scale regime in which is kept fixed (but large). Traditionally, it is the asymptotic regime which is studied, which connects the subject to other asymptotic fields of mathematics, such as dynamical systems and ergodic theory. However, for many applications (such as the study of the primes), it is the single-scale regime which is of greater importance. The two regimes are not directly equivalent, but are closely related: the single-scale theory can be usually used to derive analogous results in the asymptotic regime, and conversely the arguments in the asymptotic regime can serve as a simplified model to show the way to proceed in the single-scale regime. The analogy between the two can be made tighter by introducing the (qualitative) ultralimit regime, which is formally equivalent to the single-scale regime (except for the fact that explicitly quantitative bounds are abandoned in the ultralimit), but resembles the asymptotic regime quite closely.
We will view the equidistribution theory of polynomial orbits as a special case of Ratner’s theorem, which we will study in more generality later in this course.
For the finitary portion of the course, we will be using asymptotic notation: , , or denotes the bound for some absolute constant , and if we need to depend on additional parameters then we will indicate this by subscripts, e.g. means that for some depending only on . In the ultralimit theory we will use an analogue of asymptotic notation, which we will review later in these notes.
Starting on Monday, March 29, I will begin my graduate class for the winter quarter, entitled “Higher order Fourier analysis“. While classical Fourier analysis is concerned with correlations with linear phases such as (where ), quadratic and higher order Fourier analysis is concerned with quadratic and higher order phases such as , , etc.
In recent years, it has become clear that certain problems in additive combinatorics are naturally associated with a certain order of Fourier analysis. For instance, problems involving arithmetic progressions of length three are connected with classical Fourier analysis; problems involving progressions of length four are connected with quadratic Fourier analysis; problems involving progressions of length five are connected with cubic Fourier analysis; and so forth. The reasons for this will be discussed later in the course, but we will just give one indication of the connection here: linear phases and arithmetic progressions of length three are connected by the identity
while quadratic phases and arithmetic progressions of length four are connected by the identity
and so forth.
It turns out that in order to get a complete theory of higher order Fourier analysis, the simple polynomial phases of the type given above do not suffice. One must also consider more exotic objects such as locally polynomial phases, bracket polynomial phases (such as , and/or nilsequences (sequences arising from an orbit in a nilmanifold ). These (closely related) families of objects will be introduced later in the course.
Classical Fourier analysis revolves around the Fourier transform and the inversion formula. Unfortunately, we have not yet been able to locate similar identities in the higher order setting, but one can establish weaker results, such as higher order structure theorems and arithmetic regularity lemmas, which are sufficient for many purposes, such as proving Szemeredi’s theorem on arithmetic progressions, or my theorem with Ben Green that the primes contain arbitrarily long arithmetic progressions. These results are powered by the inverse conjecture for the Gowers norms, which is now extremely close to being fully resolved.
Our focus here will primarily be on the finitary approach to the subject, but there is also an important infinitary aspect to the theory, originally coming from ergodic theory but more recently from nonstandard analysis (or more precisely, ultralimit analysis) as well; we will touch upon these perspectives in the course, though they will not be the primary focus. If time permits, we will also present the number-theoretic applications of this machinery to counting arithmetic progressions and other linear patterns in the primes.