2 1. Higher order Fourier analysis

1.1. Equidistribution of polynomial sequences in tori

(Linear) Fourier analysis can be viewed as a tool to study an arbitrary func-

tion f on (say) the integers Z, by looking at how such a function correlates

with linear phases such as n → e(ξn), where e(x) :=

e2πix

is the funda-

mental character, and ξ ∈ 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 text we will be studying higher-order correlations, such as the

correlation of f with quadratic phases such as n →

e(ξn2),

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

N

n=1

e(αn2).

Such sums are closely related to the distribution of expressions such as

αn2 mod 1 in the unit circle T := R/Z, as n varies from 1 to N. More

generally, one is interested in the distribution of polynomials P :

Zd

→ T

of one or more variables taking values in a torus T; for instance, one might

be interested in the distribution of the quadruplet

(αn2,α(n

+

r)2,α(n

+

2r)2,α(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 qua-

dratic 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 distri-

bution 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

quadruplet1

(1A(n), 1A(n+r), 1A(n+2r), 1A(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 the previous monograph

[Ta2009]; it will also be important in this text (particularly as a source of

motivation), but the primary focus will be on finitary, and Fourier-based,

methods.

1Here 1A is the indicator function of A, defined by setting 1A(n) equal to 1 when n ∈ A and

equal to zero otherwise.