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].
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