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IV. Sequences without Arithmetic Progressions
44
So let α ≤ N be chosen, and let ω be any αth root of unity, i.e.,
α
ω 1. To estimate q m (ω), let us write it as
 
α
 
ω β  1 − C P 1  ,
 
β 1 a∈S k<m
a<m k≡β(α)
a≡β(α)
and let us note that the first inner sum

σ β 1
a∈S
a<m
a≡β(α)
counts the size of a subset of S, which therefore has P which is affine
m
m
to a subset of 0, , and so has at most f elements (where we
α α
write fàx) for fà x )).
Thus
α
β m
q m (ω) − ω f − σ β
α
β 1
α
m
+ ω β f − C P 1
α
β 1 k<m
k≡β(α)
α α $ %
m m m
≤ f − σ β + f − C P (3)

α α α

β 1 β 1
α α
m m m
f − σ β + f − C P
α α α
β 1 β 1
α
m
2αf − σ β − C P m.
α
β 1
α
If we next note that σ β is exactly the number of elements of
β 1
S which are below m and so is equal to f (n) minus the number of
elements of S which are ≥ m, we obtain
α

σ β ≥ f (n) − fàn − m) ≥ C P n − fàn − m). (4)
β 1