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Given two unit vectors ${v,w}$ in a real inner product space, one can define the correlation between these vectors to be their inner product ${\langle v, w \rangle}$, or in more geometric terms, the cosine of the angle ${\angle(v,w)}$ subtended by ${v}$ and ${w}$. By the Cauchy-Schwarz inequality, this is a quantity between ${-1}$ and ${+1}$, with the extreme positive correlation ${+1}$ occurring when ${v,w}$ are identical, the extreme negative correlation ${-1}$ occurring when ${v,w}$ are diametrically opposite, and the zero correlation ${0}$ occurring when ${v,w}$ are orthogonal. This notion is closely related to the notion of correlation between two non-constant square-integrable real-valued random variables ${X,Y}$, which is the same as the correlation between two unit vectors ${v,w}$ lying in the Hilbert space ${L^2(\Omega)}$ of square-integrable random variables, with ${v}$ being the normalisation of ${X}$ defined by subtracting off the mean ${\mathbf{E} X}$ and then dividing by the standard deviation of ${X}$, and similarly for ${w}$ and ${Y}$.

One can also define correlation for complex (Hermitian) inner product spaces by taking the real part ${\hbox{Re} \langle , \rangle}$ of the complex inner product to recover a real inner product.

While reading the (highly recommended) recent popular maths book “How not to be wrong“, by my friend and co-author Jordan Ellenberg, I came across the (important) point that correlation is not necessarily transitive: if ${X}$ correlates with ${Y}$, and ${Y}$ correlates with ${Z}$, then this does not imply that ${X}$ correlates with ${Z}$. A simple geometric example is provided by the three unit vectors

$\displaystyle u := (1,0); v := (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}); w := (0,1)$

in the Euclidean plane ${{\bf R}^2}$: ${u}$ and ${v}$ have a positive correlation of ${\frac{1}{\sqrt{2}}}$, as does ${v}$ and ${w}$, but ${u}$ and ${w}$ are not correlated with each other. Or: for a typical undergraduate course, it is generally true that good exam scores are correlated with a deep understanding of the course material, and memorising from flash cards are correlated with good exam scores, but this does not imply that memorising flash cards is correlated with deep understanding of the course material.

However, there are at least two situations in which some partial version of transitivity of correlation can be recovered. The first is in the “99%” regime in which the correlations are very close to ${1}$: if ${u,v,w}$ are unit vectors such that ${u}$ is very highly correlated with ${v}$, and ${v}$ is very highly correlated with ${w}$, then this does imply that ${u}$ is very highly correlated with ${w}$. Indeed, from the identity

$\displaystyle \| u-v \| = 2^{1/2} (1 - \langle u,v\rangle)^{1/2}$

(and similarly for ${v-w}$ and ${u-w}$) and the triangle inequality

$\displaystyle \|u-w\| \leq \|u-v\| + \|v-w\|,$

we see that

$\displaystyle (1 - \langle u,w \rangle)^{1/2} \leq (1 - \langle u,v\rangle)^{1/2} + (1 - \langle v,w\rangle)^{1/2}. \ \ \ \ \ (1)$

Thus, for instance, if ${\langle u, v \rangle \geq 1-\varepsilon}$ and ${\langle v,w \rangle \geq 1-\varepsilon}$, then ${\langle u,w \rangle \geq 1-4\varepsilon}$. This is of course closely related to (though slightly weaker than) the triangle inequality for angles:

$\displaystyle \angle(u,w) \leq \angle(u,v) + \angle(v,w).$

Remark 1 (Thanks to Andrew Granville for conversations leading to this observation.) The inequality (1) also holds for sub-unit vectors, i.e. vectors ${u,v,w}$ with ${\|u\|, \|v\|, \|w\| \leq 1}$. This comes by extending ${u,v,w}$ in directions orthogonal to all three original vectors and to each other in order to make them unit vectors, enlarging the ambient Hilbert space ${H}$ if necessary. More concretely, one can apply (1) to the unit vectors

$\displaystyle (u, \sqrt{1-\|u\|^2}, 0, 0), (v, 0, \sqrt{1-\|v\|^2}, 0), (w, 0, 0, \sqrt{1-\|w\|^2})$

in ${H \times {\bf R}^3}$.

But even in the “${1\%}$” regime in which correlations are very weak, there is still a version of transitivity of correlation, known as the van der Corput lemma, which basically asserts that if a unit vector ${v}$ is correlated with many unit vectors ${u_1,\dots,u_n}$, then many of the pairs ${u_i,u_j}$ will then be correlated with each other. Indeed, from the Cauchy-Schwarz inequality

$\displaystyle |\langle v, \sum_{i=1}^n u_i \rangle|^2 \leq \|v\|^2 \| \sum_{i=1}^n u_i \|^2$

we see that

$\displaystyle (\sum_{i=1}^n \langle v, u_i \rangle)^2 \leq \sum_{1 \leq i,j \leq n} \langle u_i, u_j \rangle. \ \ \ \ \ (2)$

Thus, for instance, if ${\langle v, u_i \rangle \geq \varepsilon}$ for at least ${\varepsilon n}$ values of ${i=1,\dots,n}$, then (after removing those indices ${i}$ for which ${\langle v, u_i \rangle < \varepsilon}$) ${\sum_{i,j} \langle u_i, u_j \rangle}$ must be at least ${\varepsilon^4 n^2}$, which implies that ${\langle u_i, u_j \rangle \geq \varepsilon^4/2}$ for at least ${\varepsilon^4 n^2/2}$ pairs ${(i,j)}$. Or as another example: if a random variable ${X}$ exhibits at least ${1\%}$ positive correlation with ${n}$ other random variables ${Y_1,\dots,Y_n}$, then if ${n > 10,000}$, at least two distinct ${Y_i,Y_j}$ must have positive correlation with each other (although this argument does not tell you which pair ${Y_i,Y_j}$ are so correlated). Thus one can view this inequality as a sort of `pigeonhole principle” for correlation.

A similar argument (multiplying each ${u_i}$ by an appropriate sign ${\pm 1}$) shows the related van der Corput inequality

$\displaystyle (\sum_{i=1}^n |\langle v, u_i \rangle|)^2 \leq \sum_{1 \leq i,j \leq n} |\langle u_i, u_j \rangle|, \ \ \ \ \ (3)$

and this inequality is also true for complex inner product spaces. (Also, the ${u_i}$ do not need to be unit vectors for this inequality to hold.)

Geometrically, the picture is this: if ${v}$ positively correlates with all of the ${u_1,\dots,u_n}$, then the ${u_1,\dots,u_n}$ are all squashed into a somewhat narrow cone centred at ${v}$. The cone is still wide enough to allow a few pairs ${u_i, u_j}$ to be orthogonal (or even negatively correlated) with each other, but (when ${n}$ is large enough) it is not wide enough to allow all of the ${u_i,u_j}$ to be so widely separated. Remarkably, the bound here does not depend on the dimension of the ambient inner product space; while increasing the number of dimensions should in principle add more “room” to the cone, this effect is counteracted by the fact that in high dimensions, almost all pairs of vectors are close to orthogonal, and the exceptional pairs that are even weakly correlated to each other become exponentially rare. (See this previous blog post for some related discussion; in particular, Lemma 2 from that post is closely related to the van der Corput inequality presented here.)

A particularly common special case of the van der Corput inequality arises when ${v}$ is a unit vector fixed by some unitary operator ${T}$, and the ${u_i}$ are shifts ${u_i = T^i u}$ of a single unit vector ${u}$. In this case, the inner products ${\langle v, u_i \rangle}$ are all equal, and we arrive at the useful van der Corput inequality

$\displaystyle |\langle v, u \rangle|^2 \leq \frac{1}{n^2} \sum_{1 \leq i,j \leq n} |\langle T^i u, T^j u \rangle|. \ \ \ \ \ (4)$

(In fact, one can even remove the absolute values from the right-hand side, by using (2) instead of (4).) Thus, to show that ${v}$ has negligible correlation with ${u}$, it suffices to show that the shifts of ${u}$ have negligible correlation with each other.

Here is a basic application of the van der Corput inequality:

Proposition 2 (Weyl equidistribution estimate) Let ${P: {\bf Z} \rightarrow {\bf R}/{\bf Z}}$ be a polynomial with at least one non-constant coefficient irrational. Then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N e( P(n) ) = 0,$

where ${e(x) := e^{2\pi i x}}$.

Note that this assertion implies the more general assertion

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N e( kP(n) ) = 0$

for any non-zero integer ${k}$ (simply by replacing ${P}$ by ${kP}$), which by the Weyl equidistribution criterion is equivalent to the sequence ${P(1), P(2),\dots}$ being asymptotically equidistributed in ${{\bf R}/{\bf Z}}$.

Proof: We induct on the degree ${d}$ of the polynomial ${P}$, which must be at least one. If ${d}$ is equal to one, the claim is easily established from the geometric series formula, so suppose that ${d>1}$ and that the claim has already been proven for ${d-1}$. If the top coefficient ${a_d}$ of ${P(n) = a_d n^d + \dots + a_0}$ is rational, say ${a_d = \frac{p}{q}}$, then by partitioning the natural numbers into residue classes modulo ${q}$, we see that the claim follows from the induction hypothesis; so we may assume that the top coefficient ${a_d}$ is irrational.

In order to use the van der Corput inequality as stated above (i.e. in the formalism of inner product spaces) we will need a non-principal ultrafilter ${p}$ (see e.g this previous blog post for basic theory of ultrafilters); we leave it as an exercise to the reader to figure out how to present the argument below without the use of ultrafilters (or similar devices, such as Banach limits). The ultrafilter ${p}$ defines an inner product ${\langle, \rangle_p}$ on bounded complex sequences ${z = (z_1,z_2,z_3,\dots)}$ by setting

$\displaystyle \langle z, w \rangle_p := \hbox{st} \lim_{N \rightarrow p} \frac{1}{N} \sum_{n=1}^N z_n \overline{w_n}.$

Strictly speaking, this inner product is only positive semi-definite rather than positive definite, but one can quotient out by the null vectors to obtain a positive-definite inner product. To establish the claim, it will suffice to show that

$\displaystyle \langle 1, e(P) \rangle_p = 0$

for every non-principal ultrafilter ${p}$.

Note that the space of bounded sequences (modulo null vectors) admits a shift ${T}$, defined by

$\displaystyle T (z_1,z_2,\dots) := (z_2,z_3,\dots).$

This shift becomes unitary once we quotient out by null vectors, and the constant sequence ${1}$ is clearly a unit vector that is invariant with respect to the shift. So by the van der Corput inequality, we have

$\displaystyle |\langle 1, e(P) \rangle_p| \leq \frac{1}{n^2} \sum_{1 \leq i,j \leq n} |\langle T^i e(P), T^j e(P) \rangle_p|$

for any ${n \geq 1}$. But we may rewrite ${\langle T^i e(P), T^j e(P) \rangle = \langle 1, e(T^j P - T^i P) \rangle_p}$. Then observe that if ${i \neq j}$, ${T^j P - T^i P}$ is a polynomial of degree ${d-1}$ whose ${d-1}$ coefficient is irrational, so by induction hypothesis we have ${\langle T^i e(P), T^j e(P) \rangle_p = 0}$ for ${i \neq j}$. For ${i=j}$ we of course have ${\langle T^i e(P), T^j e(P) \rangle_p = 1}$, and so

$\displaystyle |\langle 1, e(P) \rangle_p| \leq \frac{1}{n^2} \times n$

for any ${n}$. Letting ${n \rightarrow \infty}$, we obtain the claim. $\Box$

(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.