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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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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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