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The Basic Approximation Lemma
sions formed. So we expect that even an approximate randomness
should produce at least one arithmetic progression. 43
The precise assertion is that of the following lemma.

k
Lemma. a z +o(n), uniformly on |z| 1.
a∈S(n;Pð z C P k≤n
Remark. In terms of the great Szemer´ edi–Furstenberg result that
C P ≡ 0 (except for P P 0 ), this is a total triviality. We are proving
what in truth is an empty result. Nevertheless we are not prepared
to give the lengthy and complex proofs of this general theorem, and
so we must prove the Lemma. (We do what we can.) The proof, in
fact, is really just an elaboration of the odds and evens considerations
above.

a
Proof. The basic strategy is to estimate q n (z) z −
a∈S
k
C P z , together with all of its partial sums at every root of
k<n
unity of order up to N (N is a parameter to be chosen later). The
point is that, if we have a bound on a polynomial and its partial sums
at a point, then we inherit a bound on that polynomial throughout
an arc around that point. (Thereby we will obtain bounds for arcs
between the roots of unity which will fill up the whole circle.)
Specifically, we have the identity

m n
p(z) z p(ξ) z
p m (ζ) + , à 1)
z z
1 − ζ 1 − ξ
ζ m<è ξ
for any polynomial p of degree at most n, where the p m denote the
partial sums. (This simply records the result of the “long division.”)

From (1) we easily obtain the bound |p(z)|≤è ζ − z|
m<è
|p m (ζ)|+|p(ζ)|, and so we conclude the following:

If all the partial sums are bounded by M at ζ, the polynomial is

bounded by M(n7 + 1)throughout an arc of length 27 (2)

centered at ζ.