1.5. Inverse conjecture over finite fields 77
[SuTrVa1999] for a precursor result); the d = 1 case was treated previously
in [BlLuRu1993]. The argument here is taken from [TaZi2010], and has
a certain “cohomological” flavour (comparing cocycles with coboundaries,
determining when a closed form is exact, etc.). Indeed, the inverse theory
can be viewed as a sort of “additive combinatorics cohomology”.
Let F,V,d,f,ε be as in the theorem. We let all implied constants depend
on d, F. We use the symbol c to denote various positive constants depending
only on d. We may assume ε is sufficiently small depending on d, F, as the
claim is trivial otherwise.
The case d = 0 is easy, so we assume inductively that d 1 and that
the claim has been already proven for d 1.
The first thing to do is to make f unit magnitude. One easily verifies
the crude bound
f
2d+1
Ud+1(V )
f
L1(V )
and thus
f
L1(V )
1 O(ε).
Since |f| 1 pointwise, we conclude that
Ex∈V 1 |f(x)| = O(ε).
As such, f differs from a function
˜
f of unit magnitude by O(ε) in
L1
norm.
By replacing f with
˜
f and using the triangle inequality for the Gowers norm
(changing ε and worsening the constant c in Proposition 1.5.1 if necessary),
we may assume, without loss of generality, that |f| = 1 throughout, thus
f = e(ψ) for some ψ : V R/Z.
Since
f
2d+1
Ud+1(V )
= Eh∈V e(∂hψ)
2d
Ud(V )
we see from Markov’s inequality that
e(∂hψ)
Ud(V )
1
O(εc)
for all h in a subset H of V of density 1
O(εc).
Applying the inductive
hypothesis, we see that for each such h, we can find a polynomial φh
Poly≤d−1(V R/Z) such that
e(∂hψ) e(φh)
L1(V )
=
O(εc).
Now let h, k H. Using the cocycle identity
e(∂h+kψ) = e(∂hφ)T
he(∂kφ)
where T
h
is the shift operator T
hf(x)
:= f(x + h), we see using H¨older’s
inequality that
e(∂h+kψ) e(φhT
hφk)
L1(V )
=
O(εc).
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