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