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
≥ 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
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.
= Eh∈V e(∂hψ)
we see from Markov’s inequality that
≥ 1 −
for all h in a subset H of V of density 1 −
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)
Now let h, k ∈ H. Using the cocycle identity
e(∂h+kψ) = e(∂hφ)T
is the shift operator T
:= f(x + h), we see using H¨older’s
e(∂h+kψ) − e(φhT