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
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