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
|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
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
where U is an upper-triangular
matrix; this allows for an integration as before, using Ux · x in place of
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
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