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## C OMPOSITIO M ATHEMATICA

b

o

### 1 (1977), p. 99-111

<http://www.numdam.org/item?id=CM_1977__35_1_99_0>

© Foundation Compositio Mathematica, 1977, tous droits réservés.

L’accès aux archives de la revue « Compositio Mathematica » (http:

//http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(2)

99

ON THE SIMULTANEOUS APPROXIMATION OF

A.

### Bijlsma

Publishing

Printed in the Netherlands

1. Introduction

Let a and b be

numbers with

a ~ 1 and

Let

denote

### exp(b log a )

for some fixed branch of the

The

### problem

is to determine whether it is

### possible

that all three numbers a, b

can be well

numbers of

bounded

### degree.

We take d fixed and consider

of

numbers of

at most

### d ;

H will denote the maximum of the

of a,

### j3

and y.

After initial results of Ricci

and Franklin

Schneider

### [9]

stated that for any E &#x3E;

there exist

many

with

Bundschuh

### [2]

remarked that in Schneider’s

a condition like

### {3 ~ Q

is needed and tried to prove a theorem without such a restric- tion. His assertion is that for any E &#x3E;

there are

many

with

where

means

### loglog. However,

there is an error at the

of the

of his Satz

so that his

too, is

valid if one

makes some extra

Earlier

that

many

with

have the

(3)

and Waldschmidt

### [4] recently improved

upon the above results

that for any

there are

many

with

### 03B2 ~ Q

and

The main purpose of this paper is to show that from all these

the condition

### {3 É 0

cannot be omitted. More

the

will be

### proved:

THEOREM 1: For any

### fixed

natural number K, there exist irrational numbers a, b E

such that

many

### triples (a, (3, 03B3) of

rational numbers

where H denotes the maximum

the

a,

### {3

and y.

In the theorems cited

### above,

it would seem more natural to

### place

a restriction upon the

### given

number b than upon

For

it is

### quite

easy to see that the estimate of

### Cijsouw

and Waldschmidt holds for

if one assumes

for real

the convergents

### pnlqn

of the continued fraction

of b

### (Note

that the real numbers b for which this condition is not

are U*-numbers

III

### §3)

and thus form a set of

measure

A

### sharper

result in the same direction is

the

next theorem.

THEOREM 2:

a,

1 a

branch

the

with

0.

or

b ~ R such that

the convergents

the continued

b

there are

many

numbers

### degree

at most d with

(4)

where H denotes the maximum

the

a,

and y and

= exp

In the

### proofs

of Theorems 1 and

the

### following

notations will be

if a is an

### algebraic number, [a[ (

denotes the maximum of the absolute values of the

of a,

the

of a,

the

of a and

### den(a)

the denominator of a. We shall make use

of the relations

### den(a) h(a) and |a| ~ h(a)

+ 1.

2. Two sequences of rational numbers

LEMMA 1: Let K, À E N be

### given.

There is a sequence

### ({3nY:=l of

rational numbers in

such that

all n E N the

hold:

PROOF: Let

### {31

be any rational number in

with

### h({31)

&#x3E; À. The sequence

will be defined

chosen.

there are

many rational

E

with the

### Only finitely

many rational numbers have

bounded

exp

### (h K+2({3n)),

so there exists a

both

&#x3E; exp

### (h K+2(03B2n))

and

LEMMA 2: Let K, À EN be

and let

be a sequence

### of

rational numbers in

such that

and

are

all

n E N. Put

=

### vn/wn,

where Vn, Wn E N,

= 1.

À is

### s uffi cien tly large,

there is a sequence

### (an)1lJ=1 of

rational numbers in

such

that

all n E N the

### following

assertions hold :

(5)

PROOF: Choose

The sequence

will be defined

### inductively;

suppose a i, ..., an have

been chosen and possess the desired

### properties. By

Bertrand’s Postulate

Theorem

there is a

number Un+l with

Notice

if À is

Consider the

of the interval

Take t E

Un+l -

### 1}.

Then

therefore the width of the

### partition

D does not exceed

the interval

has a

than

### wn+ilun+i,

so that this interval contains at least one of the

### points

of D. This proves the existence of a tn+1 E

Unll -

### 1}

with

(6)

and

3. Proof of Theorem 1

1. Take À E N and let

### ({3n)~n=l

be a sequence of rational numbers in

such that

are satisfied.

=

### vn/wn,

where vn, wn E N,

### (vn, wn)

= 1. We may suppose that À is

in the sense of

Lemma

let

### (an)n-=,

be a sequence of rational numbers in

such

that

are satisfied.

=

where tn, Un E

1. Define yn : =

we have

so

Q and

= u,,- u,,- =

Therefore

II. The sequence

has the

if wi is

so x E

III. From

we see that

is a

### Cauchy

sequence; it converges

to a

### limit,

which we shall call a. Then

(7)

Theorem 186 of

### [6], implies

that a is irrational.

In the same way one can prove that

is a

### Cauchy

sequence and that its limit b satisfies

so

### that b,

too, is irrational.

IV. The function xy is

### continuously

differentiable on every

subset K of

x

### (0, 1),

so that a constant

on

### K,

can be found with

From this follows the existence of a constant

on

a and

such that

for

### sufficiently large

n. 0

It may be of interest to note that a

of Theorem 1

can be

with

less

### difficulty. Indeed,

it suffices to

construct a sequence

### ({3nr:=1

of natural numbers with the

exp

and

=

### p-2.

If b is the

limit of this sequence and

many

of natural

numbers

where H = max

and

is defined

means of the

and

function.

4. A result on

linear forms

LEMMA 3:

### Suppose

d E tBI, K a compact subset

the

not

and

branches

the

on

such that

### 11

does not take the value 0. Then

many

E K X K

numbers

### of degree

at most d have the

(8)

property that a

El 0 exists with

and

where H = max

PROOF: I.

a, y E

E

### Q,

such that the conditions of the lemma are fulfilled.

### By

CI, c2, ... we shall denote natural numbers

and

### 12;

we suppose that H is

than such a

P ut B : =

then

Define

We introduce the

function

where aÀ1z = exp

= exp

and where

are rational

### integers

to be determined later. We have

Now

T : =

and consider the

of linear

These are ST

in the

+

unknowns

### p(Ài, À2);

the coeffi- cients are

### algebraic integers

in the number field

of

### degree

at

most d2. The absolute values of the

### conjugates

of the coefficients are

less than or

is

(9)

As

Lemme 1.3.1 of

### [11] ]

states that there is a non-trivial choice for the

such that

while

II. For kEN U

we

=

suppose

for

our

choice of the

we have

This is

### proved by induction;

for k = 0 the assertion is

Now suppose that

holds for

while

Lemma 7 of

we have

Here

and

Substitution in

### (13) gives

(10)

so

from which we conclude

is

and formula

of

### [11]

states that every non-zero

has the

### property

Now

so either

in the latter case

and

= 0 for s =

,S -

t =

1. This

the

of

### (12).

III. Now let k be the

### largest

natural number with

From

it follows that

Once more

Lemma 7 of

this

and

also

remains

but from the

### maximality

of k we now get

(11)

so

whence

Conclusion:

For these values of t we have

so

### according

to 1

in the latter case

and

(12)

As the

### p(Ài,À2)

are not all zero, it follows that the coefficient matrix of the system

### (20),

which is of the Vandermonde

must be

### singular.

From this we deduce the existence of

...,

with À +

=

+

or

This

so we

H

### (and B).

D

5. Proof of Theorem 2

1. The case

R is

### trivial;

we shall therefore suppose that b is real and that its continued fraction

has the

### property

described in the theorem. Let

be a

### triple fulfilling

the conditions of the

we

to be

### greater

than a certain bound

on E,

As a ~ 0 and

### a ~ 0,

we may assume a ~ 0 and

For

chosen branches

and

of the

we have

from

0 we thus

### get li(a) ~

0. As a consequence of

### (21), (22)

and

we have

If it were the case that

Theorem 1 of

would

= 0.

II. We have

that

and

are

### linearly dependent

(13)

over the field of all

Theorem 1 of

### [1]

we find

that these numbers must also be

over 0. In other

there are

E

### Q,

not both zero, such that

+

= 0.

0 because

so

### using

Lemma 3 above we see that

H.

=

then q

### log H,

so

As q must tend to

with

we may assume

and

Satz 2.11 of

is a

say

=

of

we

### have,

for some constant c,

### (23).

D

REFERENCES

[1] A. BAKER: Linear forms in the logarithms of algebraic numbers (II). Mathematika 14 (1967) 102-107.

[2] P. BUNDSCHUH: Zum Franklin-Schneiderschen Satz. J. Reine Angew. Math. 260 (1973) 103-118.

[3] P. L. CIJSOUW: Transcendence measures of exponentials and logarithms of algebraic numbers. Compositio Math. 28 (1974) 163-178.

[4] P. L. CIJSOUW and M. WALDSCHMIDT: Linear forms and simultaneous ap-

proximations (to appear).

[5] P. FRANKLIN: A new class of transcendental numbers. Trans. Amer. Math. Soc.

42 (1937) 155-182.

[6] G. H. HARDY and E. M. WRIGHT: An introduction to the theory of numbers, 4th edition. Oxford University Press, 1960.

[7] O. PERRON: Die Lehre von den Kettenbrüchen. Band I, 3. Auflage. B. G. Teubner, Stuttgart, 1954.

[8] G. RICCI: Sul settimo problema di Hilbert. Ann. Scuola Norm. Sup. Pisa (2) 4 (1935) 341-372.

(14)

[9] TH. SCHNEIDER: Einführung in die transzendenten Zahlen. Springer-Verlag, Ber- lin, 1957.

[10] A. A.

### 0160MELEV:

A. O. Gelfond’s method in the theory of transcendental numbers (in Russian). Math. Zametki 10 (1971) 415-426. English translation: Math. Notes 10 (1971) 672-678.

[11] M. WALDSCHMIDT: Nombres transcendants. Lecture Notes in Mathematics 402.

Springer-Verlag, Berlin, 1974.

(Oblatum 27-VIII-1976 &#x26; 29-IX-1976) Mathematisch Instituut Universiteit van Amsterdam Amsterdam

The Netherlands

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