76 1. Higher order Fourier analysis
and f
L2(V )
= (

ξ∈
ˆ
V
|
ˆ(ξ)|2)1/2,
f where
ˆ
V is the space of all homomor-
phisms ξ : x ξ · x from V to R/Z, and
ˆ(ξ)
f := Ex∈V f(x)e(−ξ · x) are the
Fourier coefficients of f.)
This conjecture for larger values of d are more difficult to establish.
The d = 2 case of the theorem was established in [GrTa2008]; the low
characteristic case char(F) = d = 2 was independently and simultaneously
established in [Sa2007]. The cases d 2 in the high characteristic case
was established in two stages: first, using a modification of the Furstenberg
correspondence principle in [TaZi2010], and then using a modification of
the methods of Host and Kra [HoKr2005] and Ziegler [Zi2007] to solve
that counterpart, as done in [BeTaZi2010]; an alternate proof was also
obtained in [Sz2010c]. Finally, the low characteristic case was recently
achieved in [TaZi2011].
In the high characteristic case, we saw from Section 1.4 that one could
replace the space of non-classical polynomials Poly≤d(V R/Z) in the
above conjecture with the essentially equivalent space of classical polyno-
mials Poly≤d(V F). However, as we shall see below, this turns out not
to be the case in certain low characteristic cases (a fact first observed in
[LoMeSa2008], [GrTa2009]), for instance, if char(F) = 2 and d 3; this
is ultimately due to the existence in those cases of non-classical polynomials
which exhibit no significant correlation with classical polynomials of equal
or lesser degree. This distinction between classical and non-classical poly-
nomials appears to be a rather non-trivial obstruction to understanding the
low characteristic setting; it may be necessary to obtain a more complete
theory of non-classical polynomials in order to fully settle this issue.
The inverse conjecture has a number of consequences. For instance, it
can be used to establish the analogue of Szemer´ edi’s theorem in this setting:
Theorem 1.5.4 (Szemer´ edi’s theorem for finite fields). Let F = Fp be
a finite field, let δ 0, and let A
Fn
be such that |A|
δ|Fn|.
If
n is sufficiently large depending on p, δ, then A contains an (affine) line
{x, x + r, . . . , x + (p 1)r} for some x, r
Fn
with r = 0.
Exercise 1.5.3. Use Theorem 1.5.4 to establish the following generalisation:
with the notation as above, if k 1 and n is sufficiently large depending on
p, δ, then A contains an affine k-dimensional subspace.
We will prove this theorem in two different ways, one using a density
increment method, and the other using an energy increment method. We
discuss some other applications below the fold.
1.5.1. The 99% inverse theorem. We now prove Proposition 1.5.1. Re-
sults of this type for general d appear in [AlKaKrLiRo2003] (see also
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