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Certain estimates are fairly evident. Pàz) is a polynomial and so
π
III. The Erd˝ os–Fuchs Theorem 33
|Pàre iθ )|dθ ≤ M, (3)
−π
independent of r (0 ≤ r< 1).
π
We can also estimate the (elliptic) integral dθ
iθ
−π |1−re |
π dθ
2 iθ by the observation that if z is any complex number in
0 |1−re |
the first quadrant, then |z|≤ z + z. Thus since for 0 ≤ θ ≤ π,
1 − re iθ is in the first quadrant, ie iθ i −iθ also is, and
iθ
e −r 1−re

1 ie iθ ≤ ( + ) ie iθ . Hence
iθ
iθ
|1−re −iθ e −r e −r
|
π dθ π ie iθ

≤ ( + ) dθ
iθ
iθ e − r
0 |1 − re | 0
π
iθ
( + ) log(e − r)
0

1 + r
( + ) log −
1 − r

1 + r
π + log .
1 − r
The bound, then, is
π

dθ 1 + r
≤ 2π + 2 log . à 4)
iθ 1 − r
−π |1 − re |
π
iθ
2
The integral |A(re )| dθ is a delight. It succumbs to Parseval’s
−π
identity. This is the observation that

π
−izθ
π
ièθ
ièθ
2
| a n e | dθ a n e ¯ a m e dθ
−π −π
π
a n ¯a m e iàn −m)θ dθ
−π m,è
π

a n ¯a m e iàn −m)θ dθ
n,z −π
and these integrals all vanish except that, when n m, they are

2
equal to 2π. Hence this double sum is 2π |a n | . The derivation is
clearly valid for finite or absolutely convergent series which covers