Gelfonds method in the theory of transcendental numbers

A. OF

O.

GEL'FOND'S

TRANSCENDENTAL

A.

A.

METHOD

IN

THE

THEORY

NUMBERS

Shmelev

UDC

511

O b t a i n e d in t h i s w o r k , w i t h the h e l p of a m e t h o d of A . O. G e l ' f o n d , a r e s e v e r a l r e s u l t s on a l g e b r a i c i n d e p e n d e n c e of the v a l u e s of an e x p o n e n t i a l function at n o n a l g e b r a i c p o i n t s .

In t h i s w o r k , u s i n g a m e t h o d of Ao 0 . G e l ' f o n d [1, 2] we o b t a i n s o m e r e s u l t s on the a l g e b r a i c n o n e x p r e s s i b i l i t y of c e r t a i n t r a n s c e n d e n t a l n u m b e r s , b e l o n g i n g to a p r e s c r i b e d c o l l e c t i o n , b y any one of t h e m . L e t C, Q a n d A b e the f i e l d s of c o m p l e x , r a t i o n a l , a n d a l l a l g e b r a i c n u m b e r s ; l e t Z b e t h e r i n g of i n t e g e r s , Z[x] the r i n g of p o l y n o m i a l s in x o v e r Z. If P(x) ~: 0, P(x) E Z[x] , then n(P) is the d e g r e e of P ( x ) , a n d h(P) the h e i g h t of P ( x ) . If 71 . . . . , "/s E A , t h e n Q ( ' r l . . . . . 7 s ) is the l e a s t e x t e n s i o n o f Q c o n t a i n i n g T1 . . . . , ~/s; i f D is s o m e a l g e b r a i c f i e l d , t h e n Z(D) is i t s r i n g of i n t e g e r s and u (D) i t s d e g r e e . F o r a n y y E A , t h e n n ( 7 ) , h ( ~ ) , and b(-y) a r e , r e s p e c t i v e l y , the d e g r e e , the h e i g h t , and t h e l e a d i n g c o e f f i c i e n t , o f t h e m i n i m a l p o l y n o m i a l f o r ~/. L e t 0 ~ A, a n d l e t Q (0) be the e x t e n s i o n of 0 b y a d j o i n i n g 0 to it; w e d e f i n e the r i n g l i o f i n t e g e r s of the f i e l d Q(0) a s t h e s e t of n u m b e r s w h i c h a r e p o l y n o m i a l s in 0 w i t h c o e f f i c i e n t s f r o m Z o If 01 is a r o o t of an a l g e b r a i c e q u a t i o n of d e g r e e u w i t h c o e f f i c i e n t s f r o m 1 , t h e n O~ is o u r t e r m f o r the f i e l d Q( 0, 0 i) o r , a s i m p l e a l g e b r a i c f i e l d o f f i n i t e d e g r e e . We s h a l l c a l l n u m e r i c a l i n t e g e r s of the f i e l d 0~ t h o s e p o l y n o m i a l s o v e r Z in 0 and 0i w h o s e d e g r e e in 0 ! d o e s not e x c e e d , - 1 ; w e s h a l l u s e the t e r m d e g r e e of a n u m e r i c a l i n t e g e r i t s d e g r e e in 0, a n d i t s h e i g h t , the m a x i m u m m o d u l u s o f i t s c o e f f i c i e n t s . T h e r i n g of i n t e g e r s of the f i e l d Q~ we d e n o t e b y I~. T H E O R E M 1. L e t the n u m b e r s fit, . . . . rid E C , b e s u c h that z t, . . . , d e n t o v e r Q s u c h t h a t m d >- 2 ( m + d), and s u c h t h a t the i n e q u a l i t y

Zm E C , a r e l i n e a r l y i n d e p e n -

I ti~l -t- .. 9 "+ Id~d ] > exp ( -- ~X in x),

0<1~11 1. . . . + / l ~ [ = x, X>Xo,

(i)

w h e r e 7 > 0 is a c o n s t a n t , h o l d f o r a l l l 1. . . . . l d r Z . T h e n t h e m d n u m b e r s e f i i Z k (i = 1 . . . . . d; k = 1, . . . . m) c a n n o t b e a l g e b r a i c a l l y e x p r e s s e d b y one of t h e m . (In the c a s e w h e r e the n u m b e r s fil . . . . . rid, z 1. . . . . z m a r e r e a l , , c o n d i t i o n (1) c a n b e d r o p p e d . ) C O R O L L A R Y 1. L e t a t , a2 E A , l e t ~ b e a q u a d r a t i c i r r a t i o n a l i t y , and l e t In a 1, In a 2 b e l i n e a r l y i n d e p e n d e n t o v e r 0 * T h e n the f o u r n u m b e r s

~ , ~,

ln~at

~ r~v~,,

e

] 11~ a l ~ in a-~

c a n n o t b e a l g e b r a i c a l l y e x p r e s s e d t h r o u g h a n y one of t h e m . F o r the p r o o f , i n s e r t into T h e o r e m 1 la at

in ai

The truth of bound (1) f o l l o w s f r o m an inequality of A. O. G e l ' f o n d (see [2], p. 167, i n e q u a l i t y (117)). Moscow Scientific Research-Institute of Instruments Construction. Translated from Matematichesk i e Z a m e t k i , V o l . 10, No. 4, p p . 4 1 5 - 4 2 6 , O c t o b e r , 1971. O r i g i n a l a r t i c l e s u b m i t t e d N o v e m b e r 11, 1968.

9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. lO011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

672

C O R O L L A R Y 2. L e t fl be a q u a d r a t i c i r r a t i o n a l i t y ; l e t IDa l, lna 2 be l o g a r i t h m s of a l g e b r a i c n u m b e r s , l i n e a r l y i n d e p e n d e n t o v e r Q . T h e n a m o n g the n u m b e r s

a t l e a s t two a r e a l g e b r a i c a l l y i n d e p e n d e n t . Into the h y p o t h e s i s o f T h e o r e m 1 we i n s e r t t ~1 = ( l ~ , ~ = ~ f n a~. ~ = ~ ~/ln al, ~ = ~ ~flh-~a2, z~ = ]/-1Ta~, z 2 = ~ : z.~ = ~ ]/-h-~a~, z~ = ~ ] ( i n a ~ and u s e the G e l ' f o n d i n e q u a l i t y a l r e a d y r e f e r r e d to. C O R O L L A R Y 3. L e t /3 be a q u a d r a t i c i r r a t i o n a l i t y , l e t 0 not b e l o n g to the q u a d r a t i c f i e l d Q ([~), and l e t l n a l , lna 2 b e l o g a r i t h m s of a l g e b r a i c n u m b e r s , l i n e a r l y i n d e p e n d e n t o v e r O . T h e n the n u m b e r s ~0

~0

al, a2, a ~ a~, a~, a 2

c a n n o t be a l g e b r a i c a l l y e x p r e s s e d in t e r m s of any one o f t h e m . tn the h y p o t h e s i s of T h e o r e m 1 we put fit = l n a l , f~2 = fl ~ l n a l , f13 = lna2, /34 = f~ lna2, z i = 1, z 2 = 8 , z 3 = 0, z 4 = f~0. The b o u n d (1), a s a b o v e , f o l l o w s f r o m G e l ' f o n d ' s i n e q u a l i t y . T h e f o l l o w i n g t h e o r e m is a s t r e n g t h e n i n g of r e s u l t s of F r a n k l i n [3] and S c h n e i d e r ([4], T h e o r e m 27). F o r m u l a t i o n o f t h e s e r e s u l t s is a l s o to b e found on p a g e 36 o f the r e v i e w [5]. T H E O R E M 2. L e t a , f i e C , a ~ 0; 1, fl~/5 O ; ~, ~, ~3~-A, ~c/~ (2, m a x { n (~), n (~), n (~3)}~<~n0, w h e r e n o -> 2 is fixed; l e t H = m a x [h(~l), h(~2), h(~3)L T h e n f o r H > H0(n0, a , /3) the i n e q u a l i t y ]a - - ~ii nLI a~ - - ~ [ q- [ ~ - - ~ ] ~ exp (--ln~Hln~ lnH).

(2)

h o ! d s . [ F o r t h e c a s e ~ a ~ O and n o -> 1, i n e q u a l i t y (2) is t r u e u n d e r the a d d i t i o n a l c o n d i t i o n In h ( ~ ) / l n H > co, w h e r e co > 0 is a f i x e d c o n s t a n t . ] C O R O L L A R Y 1. L e t ~ A , h ( ~ ) = H , n ( ~ ) - < n o, w h e r e n o >-- 1 i s fixed; let P ( x l , x 2) be a p o l y n o m i a l in two i n d e p e n d e n t v a r i a b l e s w i t h c o e f f i c i e n t s f r o m Z i r r e d u c i b l e o v e r O s u c h t h a t Op/3x~ ~ 0, 0 P / ~x 2 ~ 0; l e t g be a f i x e d t r a n s c e n d e n t a I r o o t of the e q u a t i o n P ( z , zZ) = 0. T h e n the i n e q u a l i t y [~ - - ~[ < exp (--lna

It in 4 In Hi

(3)

has only a finite number of solutions. C O R O L L A R Y 2.

L e t ~0 = (~2/~/I) be i r r a t i o n a l and l e t the i n e q u a l i t y ]q0 - - ~[ < exp (--In 4 H ln~ln H)

(4)

h a v e an i n f i n i t e s e t of s o l u t i o n s in n u m b e r s ~ ~ A o f f i x e d d e g r e e , h(~) = H. T h e n t h e t h r e e n u m b e r s ~0, e ~ 1, e~2 c a n n o t b e e x p r e s s e d in t e r m s of a n y one of t h e m . C o r o l l a r i e s 1 and 2 a r e r e a d i l y p r o v e d o n r e p e a t i n g r e a s o n i n g in N. I. F e l ' d m a n ' s w o r k [6] (p. 16) u s i n g b o u n d s g i v e n in L e m m a 5 of [7] and L e m m a 6, C h a p t e r 3, o f [2]. C O R O L L A R Y 3.

L e t t h e n u m b e r s a , fl be a l g e b r a i c a l l y i n d e p e n d e n t o v e r Q , a n d l e t the i n e q u a l i t y [a .... ~[ -k [[~ - - ~[ < exp (--ln ~ H ht ~ in H)

h a v e an i n f i n i t e s e t of s o l u t i o n s in ~l, ~2 ~ A , m a x ~n (~)1, n (r

(5)

-< n 0, w h e r e no -> 1 is f i x e d ,

[I = max {h'(~), hi(~)}, Inh:(~e)/lnH > c~, w h e r e c 1 > 0 i s a c o n s t a n t . T h e n t h e n u m b e r s a , fl a n d a/~ a r e a l g e b r a i c a l l y i n d e p e n d e n t . (If a m o n g the n u m b e r s ~ f i n i t e l y m a n y a r e i r r a t i o n a l , t h e n the c o n d i t i o n in h (~2)/ln H > c 1 c a n b e d r o p p e d . ) P r o o f . S u p p o s e the n u m b e r s a, fl, a/~ w e r e a l g e b r a i c a l l y d e p e n d e n t , i . e . , t h a t t h e r e e x i s t e d an i r r e d u c i b l e p o l y n o m i a l P ( x 1, x2, x~), c o n t a i n i n g xs, with c o e f f i c i e n t s f r o m g s u c h t h a t P (a, fl, aft) = 0. S u p p o s e P0 is the d e g r e e o f t h i s p o l y n o m i a l in the c o l l e c t i o n of v a r i a b l e s ~l = n(~l), u~ = n(~2)

i p (~, ~., a ' ) ] ~ ] P ( ~ , ~, a~) - - P (~, ~e, a~)[ q- ] P (a, ~, a~', -- P (~, ~, a ~ ) [ ~ ~-

f~:~~b ~

(~' x~, a~)

ia,,~ ~OP (x~, ~, a,~)dx~

dx~'<exp (--o-i-o- i la~ H In ~ In H ) . 673

L e t ~1 = ~l,l; l e t ~t,2 . . . . . The number

~l,v 1 b e a d j o i n t f o r ~l; l e t ~2 = ~2,1; l e t ~2,2 . . . . .

~2,vi b e a d j o i n t f o r ~2.

~ ~j~= p(~,~, ~,j, a~) i s a p o l y n o m i a l in a f t o f d e g r e e n o t h i g h e r t h a n u0 ul u2 and of h e i g h t not g r e a t e r t h a n H'- ....... . U s i n g L e m m a 5 f r o m [7] and L e m m a 6 f r o m C h a p t e r 3 of [2], we g e t t h a t t h e r e e x i s t s ~:~ ~ A, n (G) < VoVaV:, h (G) H a'~o','~, s u c h t h a t

S i n c e m a x ~ ( ~ l ) , h(~2), h(~a)} -< H .~......... , and the d e g r e e s of t h e s e n u m b e r s a r e b o u n d e d , it f o l l o w s f o r H > H0 by Theorem 2 la - - ~ I + I~ - - ~ l + la:s - - GI ~ (--243v~v~v~In~ H .la~ h ~ / / ) , w h i c h c o n t r a d i c t s (6) and e s t a b l i s h e s the t r u t h of C o r o I l a r y 3. F r o m C o r o l l a r y 2 and A . O. G e l ' f o n d ' s T h e o r e m 2 ([2], p. 166) f o l l o w s a c e r t a i n s t r e n g t h e n i n g o f G e l ' f o n d ' s T h e o r e m 2, f o r m u l a t e d in T h e o r e m 3 and c o n s i s t i n g in d r o p p i n g the l i m i t a t i o n s on p a r a m e t e r s ~1 a n d ~2 ( s e e [2], p. 166, i n e q u a l i t y (114)), T H E O R E M 3. L e t the n u m b e r s ~1 and ~2 a s w e l l a s oq, a2, o~ = 1, be l i n e a r l y i n d e p e n d e n t o v e r Q~ T h e n at l e a s t one of the t e n n u m b e r s ~1, ~2, e l , ~ eC~i~k (i = 1, 2, 3; k = 1, 2) d o e s not b e l o n g to the f i e l d We note t h a t i n s t e a d o f C o r o l l a r y 2, f o r the d e r i v a t i o n o f T h e o r e m 3, w e c o u l d u s e s o m e r e s u l t s of [6]. W e a p p e n d s o m e c o r o l l a r i e s of T h e o r e m 3. C O R O L L A R Y 1. S u p p o s e 0 ~ Q is not a q u a d r a t i c i r r a t i o n a l i t y . T h e n the five n u m b e r s e 02, e 03, e 04 c a n n o t b e a l g e b r a i c a l l y e x p r e s s e d in t e r m s o f a n y one o f t h e m .

0, e 0,

C O R O L L A R Y 2, L e t ~1 ~ Q , and l e t In a l , In a2, In a 3 b e l o g a r i t h m s o f a l g e b r a i c n u m b e r s , l i n e a r l y i n d e p e n d e n t o v e r Q . T h e n the n u m b e r s ~ , a~t, a2~, a~, In a~, In a2, In a 3 c a n n o t be a l g e b r a i c a l l y e x p r e s s e d in t e r m s of a n y one of t h e m . P r o o f of T h e o r e m 1. S u p p o s e t h a t on a j o i n i n g to the f i e l d O the n u m b e r s e f i i Z k (i = 1 . . . . . d; k = 1..... m) the f i e l d O~ h a s b e e n o b t a i n e d . By a t h e o r e m of S. L a n g ([8], p . 8) a t l e a s t one of t h e s e n u m b e r s is t r a n s c e n d e n t a l . H e n c e we s h a l l a s s u m e t h a t the f i e l d O[ i s g e n e r a t e d b y the n u m b e r s 0 and 01~ W i t h o u t l o s s of g e n e r a l i t y we c a n c o n s i d e r t h a t t h e h i g h e s t c o e f f i c i e n t of the e q u a t i o n f o r 0~ i s 1 and t h a t t h e r e m a i n i n g c o e f f i c i e n t s a r e n u m b e r s o f the r i n g t7. L e t 02 . . . . . 0v b e a d j o i n t f o r 0i r e l a t i v e to the f i e l d Q (0). By a s s u m p t i o n , the m d n u m b e r s

e3iZk

(~=1,

..,d;

k=t

(7)

. . . . ,m)

c o i n c i d e with m d n u m b e r s o f the f i e l d SitT~

w h e r e S i , T i 611" (i = 1 . . . . .

(i = t , . . . , rod),

(8)

T = T~. . . T~d,

rod).

Consider the function ](z)

N "X~N C - e(~q-'"l-tca~a)z t = ~k~=0 " d-J,~ 0 ~.......d

1

(9)

w h e r e N ~ Z h e r e a n d t h r o u g h o u t t h e s e q u e l d e n o t e s a l a r g e enough p o s i t i v e n u m b e r . L e t Xi > 0 (i ~: 1, 2 . . . . ) be c o n s t a n t s n o t d e p e n d i n g on N . R e p e a t i n g c o n s i d e r a t i o n s i n t r o d u c e d in the p r o o f of T h e o r e m 1, C h a p t e r 3 [2] (page 196) we c o n c l u d e t h a t t h e r e e x i s t Ck0,k I . . . . . kd ~ Z , in a g g r e g a t e d i f f e r e n t f r o m z e r o , 0 < m a x tCk0,kl, . . . . . k d [ < exp (XlN(m+d)/m) s u c h t h a t f ( z ) = 0 a t the p o i n t s

674

z -- l,zt +. 9 .q- l,~z,~, O.<_li~[7.=Nel"],

•

•

Let g ~ C be an arbitrary number. . . . . )o

1 ~i~m.

(t0)

Define t h e s y m b o l (g) b y the r e l a t i o n (g) = m i n l g - k f

(k = 0,

In [2] (p. 180) the i n e q u a l i t y ~ L + v Ig -- k i > (g) 2-L-~ L!.

(ii)

k~p

w a s d e m o n s t r a t e d . U s i n g i n e q u a l i t i e s (1) a n d (11), w e g e t , f o r , 8'1 : f&f + (1/l&f, o -< L i tkil+ . ~ +Ikdt >0,

~,=-L~" 9

kd=_~d l h~, + . . +.

~=_~ ke~l > (~) -(N<e I~ '~-~0=_~ 9 . . _Vf.~N-Ld (k= ~~ +. ..

+

<- N (i : 1 , . . . .

d),

,~ ' ~7)~d (i2)

• 2-N-1 N! > (~)-(N+I)d 2-(N+I)d (N!)(N+I)~-Ie-~'(N+1)em N > e - 2 ~ ,,

2/2

W e u s e L e m m a 3, C h a p t e r 3 [2], s u b s t i t u t i n g into it T1 ~ y t h a t l e m m a a t l e a s t one o f t h e n u m b e r s

=

l / d , T0 : 2 / m , s = l / d , T0 = 2 / m , s : i / d ,

= 2T/d,

is d i f f e r e n t f r o m z e r o . S u p p o s e r 1 = [8(1 + d + T) 1 / m N d / m ] + 1, z 0 = ll,0Z 1 + . . . +l m,0Zm, 0 <-- Zi, 0 --< r , - - 1 , 1 ~ i --< m , i s one s u c h n u m b e r s o t h a t L (Zo) = T ~'~ ] (1~,o z~ + . . .

+ l~,~z~) ,~- 0.

t t f o l t o w s f r o m (9) a n d (13) t h a t f0(z0) ~ I 1. U s i n g (9), (10), a n d (13), we bound t h e q u a n t i t y If0(z0) t :

+ w h e r e 17 ts t h e c i r c l e t g t = N ( d + j / m ,

n: C [z[ -< 8 r , ( 1 + l z , [ + 9 9 9 + t z m ] ) .

(-, It f o l l o w s f r o m (14) t h a t

If0 (zo) I < exp (--)~sN a tn N).

(i5)

F r o m (9) a n d (13) t h e i n e q u a l i t y

s ~--!

v--I

follows, whore (16) L e t f 0 ( z 0 ) = f i , o ; c o n s i d e r the n u m b e r s f~,o = ~ i - o l ~ , 5 ~ Dr,r~0~0[' It is known t h a t f i , 0

* 0 (i = 2 . . . . .

( i = 2 . . . . . ~).

u)~ H e n c e

r (0) = I-L1 ]~,~4= 0; b e s i d e s 5~(0) is a p o l y n o m i a l in 0 w i t h c o e f f i c i e n t s f r o m Z , w h o s e h e i g h t h(~) and d e g r e e n ( ~ ) , a c c o r d i n g to (16), s a t i s f y t h e i n e q u a l i t y max {n ((D), in h (r

~ %8N (m+a)lm.

(17)

I t f o l l o w s f r o m i n e q u a l i t i e s (15) and (16) t h a t ]rb (0)[ < exp (--)~ N a in N).

(18)

To the s e q u e n c e of p o l y n o m i a l s (I,(0), o b t a i n e d f o r d i f f e r e n t N, w e now a p p l y L e m m a 7 f r o m C h a p t e r 3 of [2], c h o o s i n g a (N) = 3 ~ N ( m + d ) / m , 0(N) = (X7/9), ~ In N; w e o b t a i n t h a t 0,5=_ A w h i c h c o n t r a d i c t s t h e c h o i c e of 0. P r o o f of T h e o r e m 1 in the r e a l c a s e p r o c e e d s w i t h o u t u s i n g c o n d i t i o n (1) s i n c e i n s t e a d o f L e m m a 3, C h a p t e r 3, [2], w e a p p l y L e m m a 1 of C h a p t e r 12 of [9].

675

P r o o f of T h e o r e m 2. Suppose T h e o r e m 2 is untrue; i.e., the inequality Ia

-

h a s an infinite s e t of s o l u t i o n s .

/(z)

=

-~-t~ -- ~ai < e x p ( - i n 4 H I n s i n H )

~lt + l a ~ - - ~If

-

Put N = [ln H In 1/4 In H] and c o n s i d e r the function

Xf.

~.

C~z

a , 3[~ ==[N~In'/,N],

w h e r e Ck~,k 2 ~ Z in a g g r e g a t e d i f f e r f r o m z e r o . v~ = n (~)

(19)

(i = t, 2, 3),

(20)

M~ ~= [Nln-V,N],

Let

1,,~ = I (l~ + l # ) , 0 ~ l~ ~ ~V~

(~ = 1, 2),

(21)

~t~,l 2 being n u m b e r s obtained by r e p l a c i n g , in the q u a n t i t i e s f l l , l 2 a by ~I, aft by ~2, fl by ~3,

(22) w h e r e the n u m b e r s Bl~,l~, . . . .

n~ ~ Z and f o r ll, 12 s a t i s f y i n g (21), the bound

holds. U s i n g L e m m a 5, C h a p t e r 12, [0], Ckl,k ~ ~

Z c a n be so chosen that f o r 0 -< lI, 12 ~ L I =

N :~

7

the s u m , t a k e n in b r a c e s in E q s . (22) v a n i s h e s , and 1 ~ m a x tCkl,k 2 l< exp (5 NI). F o r this c h o i c e of Ckl,k 2 ~,~=

0, O~l~, l ~ L ~ = [

l

N~]

(23)

Since f o r 0 --< II, 12 -< N 3 (24) it follows, f o r tl, t2, s a t i s f y i n g (23), that tf/l,/2}< exp ( - ( 9 / 1 2 ) N 4 In 2 N). We bound f r o m below ~he q u a n t i t i e s ili +12~ I, 0 -< li, 12 - N3~ F r o m the inequality ik l + k2fll> Ik 1 + k 2 ~ t - f k 2 ( ~ 3 - f i ) t and the bound lk t + k 2 ~ l > 10e-N, N > N 2, c o n s t i t u t i n g the c o r o l l a r y of L e m m a 2 [10], in the light of (19), we get that tk i +k2fii> 9e-N. Suppose p = 3e -N. Applying the Hermit(an interpolation formula for I z [ -< N2 In N LI

L~

w h e r e the c o n t o u r F is the c i r c l e I ~i = N 3, and w h e r e Fll,12 is the c i r c l e t ~-ll-12fil = p. It follows f r o m (20) and (25) that (26) U s i n g the m e t h o d of [I1], we bound the i n t e g r a l o v e r Yli,l 2. L e t

R e p e a t i n g the r e a s o n i n g applied in [11] (on p. 486), we get ~ pa (Ll -- 2)! 2L~-3

whence follows max It follows f r o m (27) and (25) that

676

I-L=0 l]k==o ~ - kl - k~ < exp (2N' In N).

(27)

__L ~L~

L~

i

L~

L~

~-k,-~.l~ ; - ,

/<exPt,--Ti-N~'ln'N)

'

(28)

and finally, f o r t z l ~ N 2 In N t N a In N ) " 9v~v~va

(29)

I 2 ~ L2 = L.V2hL'/'Nt,

(30)

If (z) [ < exp ( C o n s i d e r the q u a n t i t i e s

St,,z~, ( ) ~ l . f o r t h e s e l i, l 2 we show that ~Plll2 = O~

Suppose that f o r at l e a s t one p a i r II, 12, gVll,l 2 ~ 0; then by L e m m a 2 of [10] I ~vlt,12f> e x p ( - 4 ~ l ~2~3N4) and lfll,l 2 t> ] gv/1,/2t-]~vll,12--fll,121, which gives the l o w e r bound I fz,.z~[ > exp (-- 5vlv~%N4),

(31)

c o n t r a d i c t i n g the bound (29). T h u s , f o r It, I2, s a t i s f y i n g (15) ~vll, l ~ = O. It follows f r o m (24) that ii~,,l~[~exp(---~N41rPN,,

O~l~, l~<~L~.

(32)

A g a i n we u s e the H e r m i t i a n i n t e r p o l a t i o n f o r m u l a , r e p l a c i n g L 1 by L2, the c o n t o u r r by the c o n t o u r ~1 - the c i r c l e l ~ I = N31n 1/2 N f o r ] z I -< N 3 In 2/~ N~ Bounding the modulus of the f i r s t t e r m in (25) we get

F o r the s e c o n d t e r m , we r e p e a t the r e a s o n i n g u s e d in d e r i v i n g bound (28); we get

F r o m (33) and (34) it follows, f o r lz] ~ N ~ In 2/5 N

]/(z)]<

e x p ( - - ~ 3 NalnV, N lnln N ) .

(35)

Let

it follows f r o m the definition o f f l ( z ) that i n e q u a l i t y (35) is fulfilled f o r f l ( z ) w h e r e I Zl ! -< i In a t N31n 2/s N. U s i n g L e m m a 8 of [11] we obtain that max /~l,k2 I 6',. ' ,. [ < exp ( - - t-~ N ' I n ' / . N l n l n N )

P

(36)

and, by choice of c o e f f i c i e n t s max [ Ck,,~ [>~ t. /r

(37)

T h e l a s t two bounds contradict, whence follows the truth of T h e o r e m 2 for i r r a t i o n a l ~3. K ~3 ~ O, ~3 = T1/T2 (TI, T2) = 1, then f o r Jkll + jk21 -< 1n6T2 it follows f r o m the inequality Ik 1 + k2/71 ~ tk 1 +k2~3-1tk2(/7-~3)t and f r o m the condition In h (~3)/In H > c o that tk 1 +k2/71 > 1/(2T). The e n s u i n g r e a s o n i n g b e g i n ning with Eq~ (25) c o i n c i d e s v e r b a t i m with that introduced a b o v e . We r e m a r k that r e a s o n i n g a n a l o g o u s to that u s e d in the p r o o f of T h e o r e m 2 e s t a b l i s h e s t r a n s e e n d e n t a ! i t y of the n u m b e r s ~1, if ~ and ~ a d m i t "too good" a s i m u l t a n e o u s a p p r o x i m a t i o n by a l g e b r a i c n u m b e r s . T h i s r e s u l t s f r o m s o m e s t r o n g e r known r e s u l t s of R i c c i , F r a n k l i n , P l a t o n o v (see [5], p. 36 and the b i b l i o graphy) in c a s e one of the n u m b e r s ~, 77 is not a l g e b r a i c , and the a p p r o x i m a t i n g e l e m e n t s have fixed d e g r e e . T H E O R E M 4. L e t ~, ~i ~ C be fixed and let the inequality [~ - - ~11 -~ Iq -- ~21 < exp (--ln3H In In H) h a v e an infinite s e t of solutions in ~l, ~2 ~ A ,

(38)

;

m a x tm(~l), n(~2)} ~ n0, w h e r e no -> 1 is c o n s t a n t ,

677

tt=max{h(~l), h(~)},

inH

>c,

where e > 0 is constant. (In the ease where among the numbers ~2 an infinite set is irrational, the eondition In h (~2)/ln H > c can be dropped.) Then the number is transcendental. Considered in the p r o o f is the auxiliary function

f(z)

~.M~ --M~

~i~1=0>~=0

A~.~zk,~k2z, M 1 = [N-'lnV~A:],

M 2 = [Nln'/-N],

where Aktk2 E Z in aggregate different f r o m z e r o , N = [In H In 1/~ In H]. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 90 10. Ii.

678

CITED

A . O . Gel'fond, "On algebraic independence of transcendental numbers of certain c l a s s e s , " Uspekhi Matem. Nauk, 4, No. 5 (33), 14-48 (1949). A . O . Gel'fond, T r a n s c e n d e n t a l and Algebraic Numbers [in Russian], Moscow (1952). P. Franklin, "A new class of transcendental n u m b e r s , " T r a n s , A m e r . Math. Soc., 42, 155-182 (1937). T. Schneider, Einfi~hriJng in die transzendente Zahlen, Berlin, 1957. N . I . F e l ' d m a n and A. B. Shidlovskii, "The development and p r e s e n t state of the theory of t r a n s cendental n u m b e r s , " Uspekhi Matem. Nauk, 22, No. 3 (135), 3-81 (1967). N~ I. F e l ' d m a n , "Arithmetic p r o p e r t i e s of the solutions of a transcendental equation," Vestnik Moskv. Gos. Univ., Ser. I, No. 1, 13-20 (1964). N . I . Fel'dman, "Approximation of some transcendental numbers, I" Izv. Akad. Nauk SSSR, Set. Matem., 15, 53-74 (1951). S. Lang, Introduction to Transcendental Numbers, London (1966). A . O . Gel'fond and Yu. V. Linnik, E l e m e n t a r y Methods in Analytic Number Theory [in Russian], Moscow (1962). N . I . F e l ' d m a n , "Refinement of bounds of linear f o r m s in logarithms of algebraic n u m b e r s , " Matem. Sbornik, 7__~7(119), No. 3, 423-436 (1969). N . I . F e l ' d m a n , "On approximation by algebraic numbers of logarithms of algebraic n u m b e r s , " Izv. Akad. Nm~k SSSR, Ser. Matem., 2 4 , 4 7 5 - 4 9 2 (1960).