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V. The Waring Problem
56
4
, 1 given as above. By Lemma 4, since
Proof of (1). Choose g ≥
1
−k
the length of each I a,b,N is at most 2N ,
g
N g
C 7 N 1
k
eàxn ) dx ≤ .
4 k
(b + jð N

I a,b,N n 1
Summing over all a, b, j gives the estimate
b
C 7 N g−k ≤ CN g−k
(b + jð 4
b,j
∞ ∞ 1
since 3 < ∞, and the proof is complete.
b 1 j 0 (b+jð