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 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)
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