60 1. Higher order Fourier analysis

(particularly when p = 2) and also in number theory; but they also serve as

an important simplified dyadic model for the integers. See [Ta2008, §1.6]

or [Gr2005a] for further discussion of this point.

The additive combinatorics of the integers Z, and of vector spaces V

over finite fields, are analogous, but not quite identical. For instance, the

analogue of an arithmetic progression in Z is a subspace of V . In many cases,

the finite field theory is a little bit simpler than the integer theory; for in-

stance, subspaces are closed under addition, whereas arithmetic progressions

are only “almost”

closed9

under addition in various senses. However, there

are some ways in which the integers are better behaved. For instance, be-

cause the integers can be generated by a single generator, a homomorphism

from Z to some other group G can be described by a single group element

g: n →

gn.

However, to specify a homomorphism from a vector space V

to G one would need to specify one group element for each dimension of

V . Thus we see that there is a tradeoff when passing from Z (or [N]) to a

vector space model; one gains a bounded torsion property, at the

expense10

of conceding the bounded generation property.

The starting point for this text (Section 1.1) was the study of equidis-

tribution of polynomials P : Z → R/Z from the integers to the unit cir-

cle. We now turn to the parallel theory of equidistribution of polynomials

P : V → R/Z from vector spaces over finite fields to the unit circle. Ac-

tually, for simplicity we will mostly focus on the classical case, when the

polynomials in fact take values in the

pth

roots of unity (where p is the

characteristic of the field F = Fp). As it turns out, the non-classical case

is also of importance (particularly in low characteristic), but the theory is

more diﬃcult; see [Ta2009, §1.12] for some further discussion.

1.4.1. Polynomials: basic theory. Throughout this section, V will be a

finite-dimensional vector space over a finite field F = Fp of prime order p.

Recall from Section 1.3 that a function P : V → R/Z is a function is a

polynomial of degree at most d if

∂h1 . . . ∂hd+1 P (x) = 0

for all x, h1,...,hd+1 ∈ V , where ∂hP (x) := P (x+h)−P (x). As mentioned

in previous sections, this is equivalent to the assertion that the Gowers

uniformity norm e(P )

Ud+1(V )

= 1. The space of polynomials of degree

at most d will be denoted Poly≤d(V → R/Z); it is clearly an additive

9For

instance, [N] is closed under addition approximately half of the time.

10Of 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.