114 1. Higher order Fourier analysis

we have from the triangle inequality that

En∈Z/NZg(n) ≥ δ − ε − on→∞;ε(1)

and, in particular,

En∈Z/NZg(n) ≥ δ/2

for N large enough. Similarly we have 0 ≤ g ≤ 2 (say) for N large enough.

From Roth’s theorem we conclude that

Λ(g, g, g) c(δ/4)

for N large enough. On the other hand, by the first hypothesis, the other

seven terms in

Λ(f, f, f) = Λ(g + (f − g),g + (f − g),g + (f − g))

are O(α(O(ε)) for N large enough. If ε is suﬃciently small depending on δ,

we obtain the claim.

Note that this argument in fact gives a value of cν(δ) that is essentially

the same as c(δ). Also, we see that the

u2

norm here could be replaced by

the U 2 norm, or indeed by any other quantity which is strong enough for

the control hypothesis to hold, and also weak enough for the approximation

property to hold.

So now we need to find some conditions on ν that will allow us to obtain

both the control and approximation properties. We begin with the control

property. One way to accomplish this is via a restriction estimate:

Lemma 1.7.4 (Restriction estimate implies control). Let ν be a measure.

Suppose there exists an exponent 2 q 3 such that one has the restriction

estimate

(1.54)

ˆ

f

q(Z/NZ)

≤ C

whenever f : Z/NZ → C obeys the pointwise bound |f| ≤ ν, where C is

independent of n. Then ν enjoys the control in the

u2

property from Lemma

1.7.3.

Proof. From Plancherel’s theorem, we see that (1.54) already holds if we

have |f| ≤ 1, so by the triangle inequality it also holds (with a slightly

different value of C) if |f| ≤ ν + 1.

Now suppose that |f|, |g|, |h| ≤ ν+1. From (1.53) and H¨ older’s inequality

one has

|Λ(f, g, h)| ≤ f

q−2

q(Z/NZ)

ˆ

f

3−q

∞(Z/NZ)

g

q(Z/NZ)

h

q(Z/NZ)

and thus by (1.54)

|Λ(f, g, h)| ≤

Cq

f

3−q

u2(Z/NZ)