78 1. Higher order Fourier analysis
On the other hand, φhT
hφk
is a polynomial of order d. Also, since H is so
dense, every element l of V has at least one representation of the form l = h+
k for some h, k H (indeed, out of all |V | possible representations l = h+k,
h or k can fall outside of H for at most
O(εc|V
|) of these representations).
We conclude that for every l V there exists a polynomial φl Poly≤d(V
R/Z) such that
(1.42) e(∂lψ) e(φl)
L1(V
)
=
O(εc).
The new polynomial φl supercedes the old one φl; to reflect this, we abuse
notation and write φl for φl. Applying the cocycle equation again, we see
that
(1.43) e(φh+k) e(φhT
hφk)
L1(V )
=
O(εc)
for all h, k V . Applying the rigidity of polynomials (Exercise 1.4.6), we
conclude that
φh+k = φhT
hφk
+ ch,k
for some constant ch,k R/Z. From (1.43) we in fact have ch,k =
O(εc)
for
all h, k V .
The expression ch,k is known as a 2-coboundary (see [Ta2009, §1.13]
for more discussion). To eliminate it, we use the finite characteristic to
discretise the problem as follows. First, we use the cocycle identity
p−1
j=0
e(T
jh∂hψ)
= 1
where p is the characteristic of the field. Using (1.42), we conclude that
p−1
j=0
e(T
jhφh)
1
L1(V )
=
O(εc).
On the other hand, T
jhφh
takes values in some coset of a finite subgroup C
of R/Z (depending only on p, d), by Lemma 1.4.1. We conclude that this
coset must be a shift of C by
O(εc).
Since φh itself takes values in some
coset of a finite subgroup, we conclude that there is a finite subgroup C
(depending only on p, d) such that each φh takes values in a shift of C by
O(εc).
Next, we note that we have the freedom to shift each φh by
O(εc)
(ad-
justing ch,k accordingly) without significantly affecting any of the properties
already established. Doing so, we can thus ensure that all the φh take val-
ues in C itself, which forces ch,k to do so also. But since ch,k =
O(εc),
we
conclude that ch,k = 0 for all h, k, thus φh is a perfect cocycle:
φh+k = φhT
hφk.
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