4
W. M . SCHMID T
ll^2ll
4«q\\ r

~
2
= "~
1/2"e
N-
1/2+e.
It therefor e remain s t o prov e th e Alternativ e Lemma .
3. Vinogradov' s Lemma .
Write e{x) = e 2nix.
Lemma 3A . Let 9 he an interval of length I with 0 / 1 . Let 3 = 3 + Z , z.e .
/£? 5e/ of translates of 3 6 y integers. Finally suppose that 0 8 /,ZT W 2/Z / r z s
natural. Then there is a real valued function iff(x), periodic with period 1, having
(i) 0(x ) = 0 unless x £ 3 fl72 ^
(ii) having a Fourier expansion
ifj(x) = / - S + J ! c^eimx),
772*0
|c | « / min(l, I ^ S r O
ZTW where the constant in « depends only on r.
This i s par t o f th e assertio n o f Lemm a 12 i n Vinogrado v [1947].
Proof. Le t 3 b e obtaine d fro m 3 b y cuttin g of f a subinterva l (neve r min d whethe
open o r closed ) o f lengt h V28 fro m bot h ends , an d pu t 3 ' = S f + Z . Writ e
I
I i f x i s i n th e interio r o f 3'
0 i f x i s i n th e exterio r o f 3' »
V2 i f x i s a boundar y poin t o f 3 ' .
The functio n ^
0
(*) ha s a Fourie r series :
Here
so tha t k ( 0 ) | / .
0
n
U ) = / - 8 + V c i0)e(mx).
0 A - * 772
7 7 2 * 0
C
i
0 ) =
/ n ^ O ^ - * ^ * = /
3
. e(-mx)dx,
Set rj = 8/(2r) an d defin e ^ j , . . . , ^
r
inductivel y b y
,/, (
x
)
=
J . f ^ t £ ( x + z)f e 0 = 1 r).
Then if/ (x) = 0 unles s x ha s distanc e trj fro m 3 I n particular , puttin g if/(x) =
i/f (x) , we have i/Kx ) = 0 unless x 3, o r (i). Eac h * A fo r r = 1 , . . . , r has a Fourie r expansio n
tf(x) = / - 8 + X c U)e(mx).
t *—* m
Here
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