84 1. Higher order Fourier analysis

Using Cauchy-Schwarz in x and y to eliminate the f variables, we conclude

that

|Ex,y,x

,y ∈V

θy−xθy−x θy −xθy−xχy−x(x)

χy−x (x )χy −x(x)χy

−x

(x )| 1.

Setting (h1,h2,h3,h4) to be the additive quadruple (y−x, y −x , y −x, y−x )

we obtain

|Eh1+h2=h3+h4 θh1 θh2 θh3 θh4

Ex∈V χh1 (x)χh2 (x + h1 − h4)χh3 (x)χh4 (x + h1 − h4)| 1

and the claim follows (note that for the quadruples obeying the stated lower

bound, h1,h2,h3,h4 must lie in H).

Applying this lemma to our current situation, we find many additive

quadruples (h1,h2,h3,h4) for which

|Ex∈V e(φh1 (x) + φh2 (x + h1 − h4) − φh3 (x) − φh4 (x + h1 − h4))| 1.

In particular, by the equidistribution theory in Section 1.4, the polynomial

φh1 + φh2 − φh3 − φh4 is low rank.

The above discussion is valid in any value of d ≥ 2, but is particularly

simple when d = 2, as the φh are now linear, and so φh1 + φh2 − φh3 − φh4

is now constant. Writing φh(x) = ξh · x + θh for some ξh ∈ V using the

standard dot product on V , and some (irrelevant) constant term θh ∈ F, we

conclude that

(1.44) ξh1 + ξh2 = ξh3 + ξh4

for many additive quadruples h1,h2,h3,h4.

We now have to solve an additive combinatorics problem, namely to

classify the functions h → ξh from V to V which are “1% aﬃne linear”

in the sense that the property (1.44) holds for many additive quadruples;

equivalently, the graph {(h, ξh) : h ∈ H} in V × V has high “additive

energy”, defined as the number of additive quadruples that it contains. An

obvious example of a function with this property is an aﬃne-linear function

ξh = Mh + ξ0, where M : V → V is a linear transformation and ξ0 ∈ V . As

it turns out, this is essentially the only example:

Proposition 1.5.8 (Balog-Szemer´ edi-Gowers-Freiman theorem for vector

spaces). Let H ⊂ V , and let h → ξh be a map from H to V such that (1.44)

holds for |V

|3

additive quadruples in H. Then there exists an aﬃne

function h → Mh + ξ0 such that ξh = Mh + ξ0 for |V | values of h in H.

This proposition is a consequence of standard results in additive com-

binatorics, in particular, the Balog-Szemer´ edi-Gowers lemma and Freiman’s