70 1. Higher order Fourier analysis
We write the left-hand side as
|Eh1,...,hd
∈V
e(ξω0 · R(x + h1 + · · · + hd ))
d
j=1
fj(h1,...,hd )|
where fj are bounded limit functions depending on x, hd +1,...,hD that are
independent of hj.
We can eliminate each fj term in turn by the Cauchy-Schwarz argument
used in Section 1.3, and conclude that
e(ξω0 · R)
Ud (V )
1,
and thus by the monotonicity of Gowers norms
e(ξω0 · R)
Ud−1(V
)
1
or, in other words, that the degree d 1 polynomial (x, h1,...,hd−1)
∂h1 . . . ∂hd−1 (ξω0 · R)(x) is biased. By the induction hypothesis, this polyno-
mial must be low rank.
At this point we crucially exploit the high characteristic hypothesis by
noting the Taylor expansion formula
P (y) =
1
(d 1)!
∂y−1P d
(y) + low rank errors.
The high characteristic is necessary here to invert (d−1)!. We conclude that
ξω0 · R is of low rank, but this contradicts the hypothesis on the R1,...,Rm
and the non-zero nature of ξω0 , and the claim follows.
Let x V and r = (rω)ω∈{0,1}D
Σ[D].
From the above proposition we
have an equidistribution result for a cube pinned at x:
|{(h1,...,hD) V
D
: x + ω1h1 + · · · + ωDhD Srω
for all ω {0,
1}D}|
=
pm
|Σ[D]|
+ o(1) |V
|D.
(1.40)
In fact, we can do a bit better than this, and obtain equidistribution even
after fixing a second vertex:
Exercise 1.4.17 (Equidistribution of doubly pinned cubes). Let (rω)ω∈{0,1}D

Σ[D],
let x Sr0 , and let ω {0,
1}D\{0}.
Then for all but o(|V |) ele-
ments y of Srω0 , one has
(1.41) |{(h1,...,hD) V
D
: x + ω1h1 + · · · + ωDhD Srω
for all ω {0,
1}D;
x + ω1h1 + · · · + ωDhD = y}|
= (
pm
|Σ[D]|
+ o(1))|V
|D−1.
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