120 1. Higher order Fourier analysis

was some f for which U and V were disjoint. Then, by the Hahn-Banach

theorem, there must exist some linear functional

(f1,f2) → f1,φ1

L2(Z/NZ)

+ f2,φ2

L2(Z/NZ)

which separates the two sets, in the sense that

f1,φ1

L2(Z/NZ)

+ f2,φ2

L2(Z/NZ)

c

for all (f1,f2) ∈ U, and

f1,φ1

L2(Z/NZ)

+ f2,φ2

L2(Z/NZ)

≤ c

for all

(f1,f2) ∈ V , where c is a real number.

From the form of U, we see that we must have φ1 = φ2. In partic-

ular, we may normalise φ = φ1 = φ2 to be on the boundary of B. As

all finite-dimensional spaces are reflexive, we see in that case that f2,φ

can be as large as

ε4

on V , and independently f1,φ can be as large as

En∈Z/NZ max(φ, 0). We conclude that

En∈Z/NZ max(φ, 0) +

ε4

≤ En∈Z/NZfφ.

As 0 ≤ f ≤ ν, we see that fφ ≤ ν max(φ, 0), and thus

En∈Z/NZ(ν − 1) max(φ, 0) ≥

ε4.

We now make the hypothesis that the dual function D(ν + 1) of ν + 1 is

uniformly bounded:

(1.58) D(ν + 1) ≤ C.

We remark that the linear forms condition (which we have not specified

explicitly) will give this bound with C = D(1 + 1) + o(1) =

222−1

+ o(1).

Since φ is a convex combination of functions of the form ±DF and

|F | ≤ ν + 1, this implies that φ is bounded uniformly as well: |φ| ≤ C.

Applying the Weierstrass approximation theorem to the function max(x, 0)

for |x| ≤ C (and noting that the

L1

norm of ν − 1 is O(1)) we conclude that

there exists a polynomial P : R → R (depending only on ε and C) such that

En∈Z/NZ(ν − 1)P (φ) ≥

ε4/2

(say). Breaking P into monomials, and using the pigeonhole principle, we

conclude that there exists a non-negative integer k = Oε,C(1) such that

|En∈Z/NZ(ν −

1)φk|

ε,C

1;

since φ was a convex combination of functions of the form ±DF , we thus

conclude that there exist F1,...,Fk with |F1|,..., |Fk| ≤ ν + 1 such that

|En∈Z/NZ(ν − 1)(DF1) . . . (DFk)|

ε,C

1.

We shall contradict this by fiat, making the hypothesis that

(1.59) En∈Z/NZ(ν − 1)(DF1) . . . (DFk) = oN→∞;k(1)