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.