74 1. Higher order Fourier analysis

and thus

Eh1,...,hd,h1,...,hd∈V e(T1(h1,...,hd) + T2(h1,...,hd))

=

p− arankd−1(P )−arankd−1(Q).

We make the substitution hj = hj + kj. Using the multilinearity of T2, we

can write the left-hand side as

Ek1,...,kd∈V Eh1,...,hd∈V e((T1 + T2)(h1,...,hd))

×

d

j=1

fj(h1,...,hd,k1,...,kd)

where the fj are functions bounded in magnitude by 1 that are independent

of the hj variable. Eliminating all these factors by Cauchy-Schwarz as in

Section 1.3, we can bound the above expression by

|Eh0,...,h0,h1,...,h1∈V

1 d 1 d

e(

ω∈{0,1}d

(−1)|ω|(T1

+ T2)(h1

ω1

, . . . , hd

ωd ))|1/2d

which, using the substitution hi := hi

1

−hi

0

and the multilinearity of T1 +T2,

simplifies to

|Eh1,...,hd∈V e((T1 +

T2)(h1,...,hd))|1/2d

which by definition of analytic rank is

p− arankd−1(P

+Q)/2d

,

and the claim follows.

1.5. The inverse conjecture for the Gowers norm I. The

finite field case

In Section 1.3, we saw that the number of additive patterns in a given set

was (in principle, at least) controlled by the Gowers uniformity norms of

functions associated to that set.

Such norms can be defined on any finite additive group (and also on

some other types of domains, though we will not discuss this point here).

In particular, they can be defined on the finite-dimensional vector spaces V

over a finite field F.

In this case, the Gowers norms U

d+1(V

) are closely tied to the space

Poly≤d(V → R/Z) of polynomials of degree at most d. Indeed, as noted

in Exercise 1.4.20, a function f : V → C of

L∞(V

) norm 1 has U

d+1(V

)

norm equal to 1 if and only if f = e(φ) for some φ ∈ Poly≤d(V → R/Z);

thus polynomials solve the “100% inverse problem” for the trivial inequality

f

Ud+1(V )

≤ f

L∞(V

). They are also a crucial component of the solution