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 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)
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