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(Linear) Fourier analysis can be viewed as a tool to study an arbitrary function {f} on (say) the integers {{\bf Z}}, by looking at how such a function correlates with linear phases such as {n \mapsto e(\xi n)}, where {e(x) := e^{2\pi i x}} is the fundamental character, and {\xi \in {\bf R}} is a frequency. These correlations control a number of expressions relating to {f}, such as the expected behaviour of {f} on arithmetic progressions {n, n+r, n+2r} of length three.

In this course we will be studying higher-order correlations, such as the correlation of {f} with quadratic phases such as {n \mapsto e(\xi n^2)}, as these will control the expected behaviour of {f} on more complex patterns, such as arithmetic progressions {n, n+r, n+2r, n+3r} of length four. In order to do this, we must first understand the behaviour of exponential sums such as

\displaystyle  \sum_{n=1}^N e( \alpha n^2 ).

Such sums are closely related to the distribution of expressions such as {\alpha n^2 \hbox{ mod } 1} in the unit circle {{\bf T} := {\bf R}/{\bf Z}}, as {n} varies from {1} to {N}. More generally, one is interested in the distribution of polynomials {P: {\bf Z}^d \rightarrow {\bf T}} of one or more variables taking values in a torus {{\bf T}}; for instance, one might be interested in the distribution of the quadruplet {(\alpha n^2, \alpha (n+r)^2, \alpha(n+2r)^2, \alpha(n+3r)^2)} as {n,r} both vary from {1} to {N}. 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 {(f(n), f(n+r), f(n+2r), f(n+3r))} for more general classes of functions {f}; this can lead for instance to an understanding of the distribution of arithmetic progressions of length {4} in the primes, if {f} is somehow related to the primes.

More generally, to find arithmetic progressions such as {n,n+r,n+2r,n+3r} in a set {A}, it would suffice to understand the equidistribution of the quadruplet {(1_A(n), 1_A(n+r), 1_A(n+2r), 1_A(n+3r))} in {\{0,1\}^4} as {n} and {r} 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 {N} is sent to infinity, and the (quantitative) single-scale regime in which {N} 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: {X \ll Y}, {Y \gg X}, or {X = O(Y)} denotes the bound {|X| \leq CY} for some absolute constant {C}, and if we need {C} to depend on additional parameters then we will indicate this by subscripts, e.g. {X \ll_d Y} means that {|X| \leq C_d Y} for some {C_d} depending only on {d}. In the ultralimit theory we will use an analogue of asymptotic notation, which we will review later in these notes.

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In his final lecture, Prof. Margulis talked about some of the ideas around the theory of unipotent flows on homogeneous spaces, culminating in the orbit closure, equidsitribution, and measure classification theorems of Ratner in the subject.  Margulis also discussed the application to metric theory of Diophantine approximation which was not covered in the preceding lecture.

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Ben Green and I have just uploaded to the arXiv our paper, “The Möbius function is asymptotically orthogonal to nilsequences“, which is a sequel to our earlier paper “The quantitative behaviour of polynomial orbits on nilmanifolds“, which I talked about in this post.  In this paper, we apply our previous results on quantitative equidistribution polynomial orbits in nilmanifolds to settle the Möbius and nilsequences conjecture from our earlier paper, as part of our program to detect and count solutions to linear equations in primes.  (The other major plank of that program, namely the inverse conjecture for the Gowers norm, remains partially unresolved at present.)  Roughly speaking, this conjecture asserts the asymptotic orthogonality

\displaystyle |\frac{1}{N} \sum_{n=1}^N \mu(n) f(n)| \ll_A \log^{-A} N (1)

between the Möbius function \mu(n) and any Lipschitz nilsequence f(n), by which we mean a sequence of the form f(n) = F(g^n x) for some orbit g^n x in a nilmanifold G/\Gamma, and some Lipschitz function F: G/\Gamma \to {\Bbb C} on that nilmanifold.  (The implied constant can depend on the nilmanifold and on the Lipschitz constant of F, but it is important that it be independent of the generator g of the orbit or the base point x.)  The case when f is constant is essentially the prime number theorem; the case when f is periodic is essentially the prime number theorem in arithmetic progressions.  The case when f is almost periodic (e.g. f(n) = e^{2\pi i \alpha n} for some irrational \alpha) was established by Davenport, using the method of Vinogradov.  The case when f was a 2-step nilsequence (such as the quadratic phase f(n) = e^{2\pi i \alpha n^2}; bracket quadratic phases such as f(n) = e^{2\pi \lfloor \alpha n \rfloor \beta n} can also be covered by an approximation argument, though the logarithmic decay in (1) is weakened as a consequence) was done by Ben and myself a few years ago, by a rather ad hoc adaptation of Vinogradov’s method.  By using the equidistribution theory of nilmanifolds, we were able to apply Vinogradov’s method more systematically, and in fact the proof is relatively short (20 pages), although it relies on the 64-page predecessor paper on equidistribution.  I’ll talk a little bit more about the proof after the fold.

There is an amusing way to interpret the conjecture (using the close relationship between nilsequences and bracket polynomials) as an assertion of the pseudorandomness of the Liouville function from a computational complexity perspective.    Suppose you possess a calculator with the wonderful property of being infinite precision: it can accept arbitrarily large real numbers as input, manipulate them precisely, and also store them in memory.  However, this calculator has two limitations.  Firstly, the only operations available are addition, subtraction, multiplication, integer part x \mapsto [x], fractional part x \mapsto \{x\}, memory store (into one of O(1) registers), and memory recall (from one of these O(1) registers).  In particular, there is no ability to perform division.  Secondly, the calculator only has a finite display screen, and when it shows a real number, it only shows O(1) digits before and after the decimal point.  (Thus, for instance, the real number 1234.56789 might be displayed only as \ldots 34.56\ldots.)

Now suppose you play the following game with an opponent.

  1. The opponent specifies a large integer d.
  2. You get to enter in O(1) real constants of your choice into your calculator.  These can be absolute constants such as \sqrt{2} and \pi, or they can depend on d (e.g. you can enter in 10^{-d}).
  3. The opponent randomly selects an d-digit integer n, and enters n into one of the registers of your calculator.
  4. You are allowed to perform O(1) operations on your calculator and record what is displayed on the calculator’s viewscreen.
  5. After this, you have to guess whether the opponent’s number n had an odd or even number of prime factors (i.e. you guess \lambda(n).)
  6. If you guess correctly, you win $1; otherwise, you lose $1.

For instance, using your calculator you can work out the first few digits of \{ \sqrt{2}n \lfloor \sqrt{3} n \rfloor \}, provided of course that you entered the constants \sqrt{2} and \sqrt{3} in advance.  You can also work out the leading digits of n by storing 10^{-d} in advance, and computing the first few digits of 10^{-d} n.

Our theorem is equivalent to the assertion that as d goes to infinity (keeping the O(1) constants fixed), your probability of winning this game converges to 1/2; in other words, your calculator becomes asymptotically useless to you for the purposes of guessing whether n has an odd or even number of prime factors, and you may as well just guess randomly.

[I should mention a recent result in a similar spirit by Mauduit and Rivat; in this language, their result asserts that knowing the last few digits of the digit-sum of n does not increase your odds of guessing \lambda(n) correctly.]

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This week I was in Columbus, Ohio, attending a conference on equidistribution on manifolds. I talked about my recent paper with Ben Green on the quantitative behaviour of polynomial sequences in nilmanifolds, which I have blogged about previously. During my talk (and inspired by the immediately preceding talk of Vitaly Bergelson), I stated explicitly for the first time a generalisation of the van der Corput trick which morally underlies our paper, though it is somewhat buried there as we specialised it to our application at hand (and also had to deal with various quantitative issues that made the presentation more complicated). After the talk, several people asked me for a more precise statement of this trick, so I am presenting it here, and as an application reproving an old theorem of Leon Green that gives a necessary and sufficient condition as to whether a linear sequence (g^n x)_{n=1}^\infty on a nilmanifold G/\Gamma is equidistributed, which generalises the famous theorem of Weyl on equidistribution of polynomials.

UPDATE, Feb 2013: It has been pointed out to me by Pavel Zorin that this argument does not fully recover the theorem of Leon Green; to cover all cases, one needs the more complicated van der Corput argument in our paper.

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In this final lecture, we establish a Ratner-type theorem for actions of the special linear group SL_2({\Bbb R}) on homogeneous spaces. More precisely, we show:

Theorem 1. Let G be a Lie group, let \Gamma < G be a discrete subgroup, and let H \leq G be a subgroup isomorphic to SL_2({\Bbb R}). Let \mu be an H-invariant probability measure on G/\Gamma which is ergodic with respect to H (i.e. all H-invariant sets either have full measure or zero measure). Then \mu is homogeneous in the sense that there exists a closed connected subgroup H \leq L \leq G and a closed orbit Lx \subset G/\Gamma such that \mu is L-invariant and supported on Lx.

This result is a special case of a more general theorem of Ratner, which addresses the case when H is generated by elements which act unipotently on the Lie algebra {\mathfrak g} by conjugation, and when G/\Gamma has finite volume. To prove this theorem we shall follow an argument of Einsiedler, which uses many of the same ingredients used in Ratner’s arguments but in a simplified setting (in particular, taking advantage of the fact that H is semisimple with no non-trivial compact factors). These arguments have since been extended and made quantitative by Einsiedler, Margulis, and Venkatesh.
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The last two lectures of this course will be on Ratner’s theorems on equidistribution of orbits on homogeneous spaces. Due to lack of time, I will not be able to cover all the material here that I had originally planned; in particular, for an introduction to this family of results, and its connections with number theory, I will have to refer readers to my previous blog post on these theorems. In this course, I will discuss two special cases of Ratner-type theorems. In this lecture, I will talk about Ratner-type theorems for discrete actions (of the integers {\Bbb Z}) on nilmanifolds; this case is much simpler than the general case, because there is a simple criterion in the nilmanifold case to test whether any given orbit is equidistributed or not. Ben Green and I had need recently to develop quantitative versions of such theorems for a number-theoretic application. In the next and final lecture of this course, I will discuss Ratner-type theorems for actions of SL_2({\Bbb R}), which is simpler in a different way (due to the semisimplicity of SL_2({\Bbb R}), and lack of compact factors).

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In this lecture, we move away from recurrence, and instead focus on the structure of topological dynamical systems. One remarkable feature of this subject is that starting from fairly “soft” notions of structure, such as topological structure, one can extract much more “hard” or “rigid” notions of structure, such as geometric or algebraic structure. The key concept needed to capture this structure is that of an isometric system, or more generally an isometric extension, which we shall discuss in this lecture. As an application of this theory we characterise the distribution of polynomial sequences in torii (a baby case of a variant of Ratner’s theorem due to (Leon) Green, which we will cover later in this course).

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In this lecture, I define the basic notion of a dynamical system (as well as the more structured notions of a topological dynamical system and a measure-preserving system), and describe the main topics we will cover in this course.

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While working on my recent paper with Ben Green, I was introduced to the beautiful theorems of Marina Ratner on unipotent flows on homogeneous spaces, and their application to questions in number theory, such as the Oppenheim conjecture (first solved by Margulis, by establishing what can retrospectively be viewed as a special case of Ratner’s theorems). This is a subject that I am still only just beginning to learn, but hope to understand better in the future, especially given that quantitative analogues of Ratner’s theorems should exist, and should have even more applications to number theory (see for instance this recent paper of Einsiedler, Margulis, and Venkatesh). In this post, I will try to describe some of the background for this theorem and its connection with the Oppenheim conjecture; I will not discuss the proof at all, largely because I have not fully understood it myself yet. For a nice introduction to these issues, I recommend Dave Morris’ recent book on the subject (and this post here is drawn in large part from that book).

Ratner’s theorem takes place on a homogeneous space. Informally, a homogeneous space is a space X which looks “the same” when viewed from any point on that space. For instance, a sphere S^2 is a homogeneous space, but the surface of a cube is not (the cube looks different when viewed from a corner than from a point on an edge or on a face). More formally, a homogeneous space is a space X equipped with an action (g,x) \mapsto gx of a group G of symmetries which is transitive: given any two points x, y on the space, there is at least one symmetry g that moves x to y, thus y=gx. (For instance the cube has several symmetries, but not enough to be transitive; in contrast, the sphere S^2 has the transitive action of the special orthogonal group SO(3) as its symmetry group.) It is not hard to see that a homogeneous space X can always be identified (as a set with an action of G) with a quotient G/\Gamma := \{ g\Gamma: g \in G \}, where \Gamma is a subgroup of G; indeed, one can take \Gamma to be the stabiliser \Gamma := \{ g \in G: gx = x \} of an arbitrarily chosen point x in X, and then identify g\Gamma with g\Gamma x = gx. For instance, the sphere S^2 has an obvious action of the special orthogonal group SO(3), and the stabiliser of (say) the north pole can be identified with SO(2), so that the sphere can be identified with SO(3)/SO(2). More generally, any Riemannian manifold of constant curvature is a homogeneous space; for instance, an m-dimensional torus can be identified with {\Bbb R}^m/{\Bbb Z}^m, while a surface X of constant negative curvature can be identified with SL(2,{\Bbb R})/\Gamma for some subgroup \Gamma of SL(2,{\Bbb R}) (e.g. the hyperbolic plane {\Bbb H} is isomorphic to SL(2,{\Bbb R})/SO(2)). Furthermore, the cosphere bundle S^* X of X – the space of unit (co)tangent vectors on X – is also a homogeneous space with structure group SL(2,{\Bbb R}). (For instance, the cosphere bundle S^* {\Bbb H} of the hyperbolic plane {\Bbb H} is isomorphic to SL(2,{\Bbb R}) / \{ +1, -1 \}.)

For the purposes of Ratner’s theorem, we only consider homogeneous spaces X in which the symmetry group G is a connected finite-dimensional Lie group, and X is finite volume (or more precisely, it has a finite non-trivial G-invariant measure). Every compact homogeneous space is finite volume, but not conversely; for instance the modular curve SL(2,{\Bbb R})/SL(2,{\Bbb Z}) is finite volume but not compact (it has a cusp). (The modular curve has two real dimensions, but just one complex dimension, hence the term “curve”; rather confusingly, it is also referred to as the “modular surface”. As for the term “modular”, observe that the moduli space of unimodular lattices in {\Bbb R}^2 has an obvious action of SL(2,{\Bbb R}), with the stabiliser of {\Bbb Z}^2 being SL(2,{\Bbb Z}), and so this moduli space can be identified with the modular curve.)

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Ben Green and I have just uploaded our paper “The quantitative behaviour of polynomial orbits on nilmanifolds” to the arXiv (and shortly to be submitted to a journal, once a companion paper is finished). This paper grew out of our efforts to prove the Möbius and Nilsequences conjecture MN(s) from our earlier paper, which has applications to counting various linear patterns in primes (Dickson’s conjecture). These efforts were successful – as the companion paper will reveal – but it turned out that in order to establish this number-theoretic conjecture, we had to first establish a purely dynamical quantitative result about polynomial sequences in nilmanifolds, very much in the spirit of the celebrated theorems of Marina Ratner on unipotent flows; I plan to discuss her theorems in more detail in a followup post to this one.In this post I will not discuss the number-theoretic applications or the connections with Ratner’s theorem, and instead describe our result from a slightly different viewpoint, starting from some very simple examples and gradually moving to the general situation considered in our paper.

To begin with, consider a infinite linear sequence (n \alpha + \beta)_{n \in {\Bbb N}} in the unit circle {\Bbb R}/{\Bbb Z}, where \alpha, \beta \in {\Bbb R}/{\Bbb Z}. (One can think of this sequence as the orbit of \beta under the action of the shift operator T: x \mapsto x +\alpha on the unit circle.) This sequence can do one of two things:

  1. If \alpha is rational, then the sequence (n \alpha + \beta)_{n \in {\Bbb N}} is periodic and thus only takes on finitely many values.
  2. If \alpha is irrational, then the sequence (n \alpha + \beta)_{n \in {\Bbb N}} is dense in {\Bbb R}/{\Bbb Z}. In fact, it is not just dense, it is equidistributed, or equivalently that

    \displaystyle\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N F( n \alpha + \beta ) = \int_{{\Bbb R}/{\Bbb Z}} F

    for all continuous functions F: {\Bbb R}/{\Bbb Z} \to {\Bbb C}. This statement is known as the equidistribution theorem.

We thus see that infinite linear sequences exhibit a sharp dichotomy in behaviour between periodicity and equidistribution; intermediate scenarios, such as concentration on a fractal set (such as a Cantor set), do not occur with linear sequences. This dichotomy between structure and randomness is in stark contrast to exponential sequences such as ( 2^n \alpha)_{n \in {\Bbb N}}, which can exhibit an extremely wide spectrum of behaviours. For instance, the question of whether (10^n \pi)_{n \in {\Bbb N}} is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether \pi is normal base 10.

Intermediate between linear sequences and exponential sequences are polynomial sequences (P(n))_{n \in {\Bbb N}}, where P is a polynomial with coefficients in {\Bbb R}/{\Bbb Z}. A famous theorem of Weyl asserts that infinite polynomial sequences enjoy the same dichotomy as their linear counterparts, namely that they are either periodic (which occurs when all non-constant coefficients are rational) or equidistributed (which occurs when at least one non-constant coefficient is irrational). Thus for instance the fractional parts \{ \sqrt{2}n^2\} of \sqrt{2} n^2 are equidistributed modulo 1. This theorem is proven by Fourier analysis combined with non-trivial bounds on Weyl sums.

For our applications, we are interested in strengthening these results in two directions. Firstly, we wish to generalise from polynomial sequences in the circle {\Bbb R}/{\Bbb Z} to polynomial sequences (g(n)\Gamma)_{n \in {\Bbb N}} in other homogeneous spaces, in particular nilmanifolds. Secondly, we need quantitative equidistribution results for finite orbits (g(n)\Gamma)_{1 \leq n \leq N} rather than qualitative equidistribution for infinite orbits (g(n)\Gamma)_{n \in {\Bbb N}}.

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