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).