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III. The Erd˝ os–Fuchs Theorem
36
or
∞
C
2 n α
A (z) + (1 − zð a n z , a n O(n ). (12)
1 − z
n 0
Of course we may assume throughout that α< 1. Thereby (12)
2n
2 −α−1
yieldstheboundM(1−r ) for a n r ,sothatweeasilyachieve
our first goal namely,
C
2
A(r )> √ , C > 0. à 13)
1 − r 2
2
As for the other goal, the Parseval upper bound on A(r ), again
2
we wish to exploit the fact that A (z) is “near” C , but this takes
1−z
some doing. From the look of (12) unlike (1), this “nearness” seems
n
to occur only where (1 − zð a n z is relatively small, that is, only
in a neighborhood of z 1. We must “enhance” this locale if we are
to expect anything from the integration, and we do so by multiplying
by a function whose “heft” or largeness is all near z 1. A handy
2
such multiplier for us is the function S (z) where
2
S(z) 1 + z + z + ··· + z N−1 , N large. à 14)
2
The multiplication of S (z) by (12) yields
2
CS (z)
2 N n
[S(z)A(z)] + (1 − z )S(z) a n z , à 15)
1 − z
which gives
CN 2
2 n
|S(z)A(z)| ≤ + 2|S(z) a n z |, à 16)
|1 − z|
and integration leads to
π

2
iθ
iθ
|S(re )A(re )| dθ
−π
π dθ

≤ CN 2 (17)
iθ
−π |1 − re |
π

iθ n
iθ
+ 2 |S(re ) a n (re ) |dθ.
−π