1.5. Inverse conjecture over finite fields 87

in the language of ergodic theory. However, in order not to have to introduce

too much additional material, we will describe the arguments here in the case

d = 3 without explicitly using ergodic theory notation. To do this, though,

we will have to sacrifice a lot of rigour and only work with some illustrative

special cases rather than the general case, and also use somewhat vague

terminology (e.g. “general position” or “low rank”).

To simplify things further, we will establish the U

3

inverse theorem

only for a special type of function, namely a quartic

phase14

e(φ), where

φ: V → F is a classical polynomial of degree 4. The claim to show then

is that if e(φ)

U 3(V )

1, then e(φ) correlates with a cubic phase. In the

high characteristic case p 4, this result can be handled by equidistribution

theory. Indeed, since

e(φ)

8

U 3(V )

= Ex,h1,h2,h3,h4 e(∂h1 ∂h2 ∂h3 ∂h4 φ(x)),

that theory tells us that the quartic polynomial

(x, h1,h2,h3,h4) → ∂h1 ∂h2 ∂h3 ∂h4 φ(x)

is low rank. On the other hand, in high characteristic one has the Taylor

expansion

φ(x) =

1

4!

∂x∂x∂x∂xφ(0) + Q(x)

for some cubic function Q (as can be seen for instance by decomposing into

monomials). From this we easily conclude that φ itself has low rank (i.e.

it is a function of boundedly many cubic (or lower degree) polynomials), at

which point it is easy to see from Fourier analysis that e(φ) will correlate

with the exponential of a polynomial of degree at most 3.

Now we present a different argument that relies slightly less on the quar-

tic nature of φ; it is a substantially more diﬃcult argument, and we will skip

some steps here to simplify the exposition, but the argument happens to ex-

tend to more general situations. As e(φ)

U 3

1, we have Δhe(φ)

U 2

1

for many h, thus by the inverse U

2

theorem, Δhe(φ) = e(∂hφ) correlates with

a quadratic phase. Using equidistribution theory, we conclude that the cubic

polynomial ∂hφ is low rank.

At present, the low rank property for ∂hφ is only true for many h. But

from the cocycle identity

(1.46) ∂h+kφ = ∂hφ + T

h∂kφ,

we see that if ∂hφ and ∂kφ are both low rank, then so is ∂h+kφ; thus the

property of ∂hφ being low rank is in some sense preserved by addition. Using

this and a bit of additive combinatorics, one can conclude that ∂hφ is low

14A good example to keep in mind is the symmetric polynomial phase e(S2/2) from Section

1.5.2, though one has to take some care with this example due to the low characteristic.