90 1. Higher order Fourier analysis

where Qh,k := Qh+k − Qh − T

hQk

is a quadratic polynomial. With any two

of Lh,Lk,Lh+k in general position, one can then conclude (using equidis-

tribution theory) that Fh , Fk , Fh+k are quadratic polynomials. Thus ∂hφ

is quadratic, and φ is cubic as before. This completes the heuristic discus-

sion of various extreme model cases; the general case is handled by a rather

complicated combination of all of these special case methods, and is best

performed16

in the framework of ergodic theory; see [BeTaZi2010]. The

various functional equations for these vertical derivatives were first intro-

duced by Conze and Lesigne [CoLe1984].

1.5.4. Consequences of the inverse conjecture for the Gowers norm.

We now discuss briefly some of the consequences of the inverse conjecture for

the Gowers norm, beginning with Szemer´ edi’s theorem in vector fields (The-

orem 1.5.4). We will use the density increment

method17.

Let A ⊂ V =

Fn

be a set of density at least δ containing no lines. This implies that the

p-linear form

Λ(1A,..., 1A) := Ex,r∈Fn 1A(x) . . . 1A(x + (p − 1)r)

has size o(1). On the other hand, as this pattern has complexity p − 2, we

see from Section 1.3 that one has the bound

|Λ(f0,...,fp−1)| ≤ sup

0≤j≤p−1

fj

Up−1(V )

whenever f0,...,fp−1 are bounded in magnitude by 1. Splitting 1A = δ +

(1A − δ), we conclude that

Λ(1A,..., 1A) =

δp

+ Op( 1A − δ

Up−1(V

))

and thus (for n large enough)

1A − δ

Up−1(V ) p,δ

1.

Applying Theorem 1.5.3, we find that there exists a polynomial φ of degree

at most p − 2 such that

|1A − δ, e(φ)|

p,δ

1.

To proceed we need the following analogue of Proposition 1.2.6:

Exercise 1.5.6 (Fragmenting a polynomial into subspaces). Let φ:

Fn

→ F

be a classical polynomial of degree d p. Show that one can partition V

into aﬃne subspaces W of dimension at least n (n, d, p), where n → ∞ as

n → ∞ for fixed d, p, such that φ is constant on each W . (Hint: Induct

16In

particular, the idea of extracting out the coeﬃcient of a key polynomial, such as the

coeﬃcient Fh(Lh) of Q, is best captured by the ergodic theory concept of vertical differentiation.

Again, see [BeTaZi2010] for details.

17An

energy increment argument is also possible, but is more complicated; see [GrTa2010b].