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IV. Sequences without Arithmetic Progressions
46
and equality is allowed). We show that
C 3
2
2
A(n;Pð P n + o(n ). (8)
2

a
The proof is by contour integration. If we abbreviate z
a∈S
−2
g(z), then we recognize A as the constant term in g(z)g(z)g(z ),
and so we may write

1 2 −2 dz
A g (zψ(z ) . à 9)
2πi |z| 1 z
k
Now writing G(z) z , g(z) C P G(z) + q(z) (where q
k<n
is “small” by the lemma). If we substitute this in (9), we obtain

1 dz
−2
2
C 3 G (z)G(z )
P
2πi |z| 1 z
plus seven other integrals. Each of these other integrals is the product
of three functions, each a G or a q, and at least one of them is a q.By
our lemma, then, we may estimate each of these seven integrals by
o(n) times an integral of the product of two functions. Both of these
functions are either a |G| or a |q|. As such each is estimable by the
Schwarz inequality, Parseval equality techniques. The final estimate
√
2
for each of these seven integrals, therefore, is o(n) nn o(n ),
and so (9) gives

1 dz
2
−2
2
A C 3 G (z)G(z ) + o(n ). (10)
P
2πi |z| 1 z
But reading (9) for the property P 0 shows that this integral is
simply A(n;P 0 ) and it is a simple exercise to show that A(n;P 0 ),
the number of triples below n which are in arithmetic progression,
n 2
is exactly . Indeed, then (10) reduces to (8). Q.E.D.
2
All of our discussion thus far has been quite general and is valid
for arbitrary affine properties. We finally become specific by letting
P P A , and we easily deduce the following:

0.
Theorem (Roth). C P A