82 1. Higher order Fourier analysis
Now let φ be a classical polynomial of degree less than
2d−1.
By Lemma
1.5.5, we can partition F2
n
into affine coordinate subspaces W of dimension
at least ωd(n) such that φ is a linear combination of S0,...,S2d−1−1 on each
such subspace. By the pigeonhole principle, we thus can find such a W such
that
|f, e(φ) L2(Fn)|
2
|f, e(φ)
L2(W
)|.
On the other hand, from Exercise 1.5.4, the function φ on W depends only
on L mod
2d−1.
Now, as dim(W ) ∞, the function L mod
2d
(which is es-
sentially the distribution function of a simple random walk of length dim(V )
on
Z/2dZ)
becomes equidistributed; in particular, for any a
Z/2dZ,
the
function f will take the values
e(a/2d)
and
−e(a/2d)
with asymptotically
equal frequency on W , whilst φ remains unchanged. As such we see that
|f, e(φ)
L2(W
)
| 0 as dim(W ) ∞, and thus as n ∞, and the claim
follows.
Exercise 1.5.5. With the same setup as the previous proposition, show that
e(S2d−1 /2)
Ud+1(Fn)
2
1, but that
e(S2d−1
/2),e(φ)
L2(Fn)
2
= on→∞;d(1)
for all classical polynomials φ of degree less than
2d−1.
1.5.3. The 1% inverse theorem: sketches of a proof. The proof of
Theorem 1.5.3 is rather difficult once d 2; even the d = 2 case is not
particularly easy. However, the arguments still have the same cohomological
flavour encountered in the 99% theory. We will not give full proofs of this
theorem here, but indicate some of the main ideas.
We begin by discussing (quite non-rigorously) the significantly simpler
(but still non-trivial) d = 2 case, under the assumption of odd characteris-
tic, in which case we can use the arguments from [Go1998], [GrTa2008].
Unsurprisingly, we will take advantage of the d = 1 case of the theorem as
an induction hypothesis.
Let V =
Fn
for some field F of characteristic greater than 2, and let
f be a function with f
L∞(V
)
1 and f
U 3(V )
1. We would like to
show that f correlates with a quadratic phase function e(φ) (due to the
characteristic hypothesis, we may take φ to be classical), in the sense that
|f, e(φ)
L2(V
)| 1.
We expand f
8
U 3(V )
as Eh∈V Δhf
4
U 2(V )
. By the pigeonhole principle,
we conclude that
Δhf
U
2(V
)
1
for “many” h V , where by “many” we mean “a proportion of 1”.
Applying the U
2
inverse theorem, we conclude that for many h, that there
exists a linear polynomial φh : V F (which we may as well take to be
classical) such that
|Δhf, e(φh)
L2(V
)| 1.
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