86 1. Higher order Fourier analysis

Thus there exists z such that

|Ex,y∈V f(z − y)f (x)f(z − x)f (y)e(M(z − x − y) · (x − y))| 1.

We collect all terms that depend only on x (and z) or only on y (and z) to

obtain

|Ex,y∈V fz,1(x)fz,2(y)e(Mx · y − My · x)| 1

for some bounded functions fz,1,fz,2. Eliminating these functions by two

applications of Cauchy-Schwarz, we obtain

|Ex,y,x

,y ∈V

e(M(x − x ) · (y − y ) − M(y − y ) · (x − x ))| 1

or, on making the change of variables a := x − x , b := y − y ,

|Ea,b∈V e(Ma · b − Mb · a)| 1.

Using equidistribution theory, this means that the quadratic form (a, b) →

Ma·b−Mb·a is low rank, which easily implies that M is virtually symmetric.

Remark 1.5.9. In [Sa2007] a variant of this argument was introduced

to deal with the even characteristic case. The key new idea is to split

the matrix of M into its diagonal component, plus the component that

vanishes on the diagonal. The latter component can made (virtually) (anti-)

symmetric and thus expressible as U + U

T

where U is an upper-triangular

matrix; this allows for an integration as before, using Ux · x in place of

1

2

Mx · x. In characteristic two, the diagonal contribution to Mx · x is linear

in x and can be easily handled by passing to a codimension one subspace.

See [Sa2007] for details.

Now we turn to the general d case. In principle, the above argument

should still work, say for d = 3. The main sticking point is that instead of

dealing with a vector-valued function h → ξh that is approximately linear in

the sense that (1.44) holds for many additive quadruples, in the d = 3 case

one is now faced with a matrix-valued function h → Mh with the property

that

Mh1 + Mh2 = Mh3 + Mh4 + Lh1,h2,h3,h4

for many additive quadruples h1,h2,h3,h4, where the matrix Lh1,h2,h3,h4 has

bounded rank. With our current level of additive combinatorics technology,

we are not able to deal properly with this bounded rank error (the main

diﬃculty being that the set of low rank matrices has no good “doubling”

properties). Because of this obstruction, no generalisation of the above

arguments to higher d has been found.

There is, however, another approach, based ultimately on the ergodic

theory work of Host and Kra [HoKr2005] and of Ziegler [Zi2007], that

can handle the general d case, which was worked out in [TaZi2010] and

[BeTaZi2010]. It turns out that it is convenient to phrase these arguments