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III. The Erd˝ os–Fuchs Theorem
32

1
Very deep arguments have even improved this to o
n 2/3 , for ex-

ample, and the conjecture is that it is actually O 3 1 for every
n 4 −1
1> 0. On the other hand, further difficult arguments show that it is

not O 3 1 .
n 4 +1
Now all of these arguments were made for the very special case
of A the perfect squares. What a surprise then, when Erd˝ os and
Fuchs showed, by simple analytic number theory, the following:

Theorem. For any set A, r(0)+r(1)+r(2)+···+r(n) C + O 1 is
n+1 3 +1
n 4
impossible unless C 0.

This will be proved in the current chapter, but first an appetizer.
We prove that r − (n) can’t eventually be constant.
So let us assume that

C
2 2
A (z) − A(z ) Pàz) + , à 1)
1 − z
P is a polynomial, and C is a positive constant. Now look for a con-
tradiction. The simple device of letting z → (−1) which worked
+
so nicely for the r + problem, leads nowhere here. The exercises in
Chapter I were, after all, hand picked for their simplicity and involved
only the lightest touch of analysis. Here we encounter a slightly heav-
ier dose. We proceed, namely, by integrating the modulus around a
circle. From (1), we obtain, for 0 ≤ r< 1,

π

iθ
2
|A (re )|dθ
−π
π π

2 2iθ
≤ |A(r e )|dθ + |Pàre iθ )|dθ (2)
−π −π
π dθ

+ C .
iθ
−π |1 − re |