118 1. Higher order Fourier analysis
If we make the hypothesis
(1.56) Ea,c,c ν(a c)ν(a c ) = 1 + o(1)
(which is a variant of (1.55), as can be seen by expanding out using Fourier
analysis), followed by Cauchy-Schwarz, we can bound this by
(1 + o(1))
(
Ea,c,c ν(a c)ν(a c )|Ebf(b + 2c)f(b + 2c )ν(−2a
b)|2
)1/4
.
We expand this out as
(1 + o(1))|Ea,b,b
,c,c
f(b + 2c)f(b + 2c)f(b + 2c )f(b + 2c )F (b, b , c, c
)|1/4
where
F (b, b , c, c ) := Eaν(a c)ν(a c )ν(−2a b)ν(−2a b ).
If the F factor could be replaced by 1, then the expression inside the absolute
values would just be f
4
U 2(Z/NZ)
, which is what we wanted. Applying the
triangle inequality and bounding f by ν, we can thus bound the previous
expression by
(
f
U 2(Z/NZ)
+ Ea,b,b
,c,c
ν(b + 2c)ν(b + 2c)ν(b + 2c )ν(b + 2c )
|F (b, b , c, c ) 1|
)1/4
.
If we make the hypotheses
(1.57) Ea,b,b
,c,c
ν(b+2c)ν(b +2c)ν(b+2c )ν(b +2c )F (b, b , c, c
)i
= 1+o(1)
for i = 0, 1, 2, then another application of Cauchy-Schwarz gives
Ea,b,b
,c,c
ν(b + 2c)ν(b + 2c)ν(b + 2c )ν(b + 2c )|F (b, b , c, c ) 1| = o(1),
so we have obtained the control in U
2
hypothesis (at least for f, and as-
suming boundedness by ν and ν +1 assuming the conditions (1.56), (1.57)).
We refer to such conditions (involving the product of ν evaluated at distinct
linear forms on the left-hand side, and a 1 + o(1) on the right-hand side) as
linear forms conditions. Generalising to the case of functions bounded by
ν + 1, and permuting f, g, h, we can soon obtain the following result (stated
somewhat informally):
Lemma 1.7.6 (Generalised von Neumann theorem). If ν obeys a certain
finite list of linear forms conditions, then the control by U
2
hypothesis in
Lemma 1.7.3 holds.
Now we turn to the approximation in U
2
property. It is possible to es-
tablish this approximation property by an energy increment method, anal-
ogous to the energy increment proof of Roth’s theorem in Section 1.2; see
[GrTa2006] for details. However, this argument turns out to be rather
complicated. We give here a simpler approach based on duality (and more
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