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On the simultaneous approximation of <span class="mathjax-formula">$a$</span>, <span class="mathjax-formula">$b$</span> and <span class="mathjax-formula">$a^b$</span>

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C OMPOSITIO M ATHEMATICA

A. B IJLSMA

On the simultaneous approximation of a, b and a

b

Compositio Mathematica, tome 35, n

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.

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

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(2)

99

ON THE SIMULTANEOUS APPROXIMATION OF

a, b AND ab

A.

Bijlsma

Publishing

Printed in the Netherlands

1. Introduction

Let a and b be

complex

numbers with

a ~ 0,

a ~ 1 and

b ~ Q.

Let

ab

denote

exp(b log a )

for some fixed branch of the

logarithm.

The

problem

is to determine whether it is

possible

that all three numbers a, b

and a’

can be well

approximated by algebraic

numbers of

bounded

degree.

We take d fixed and consider

triples (a, (3, y)

of

algebraic

numbers of

degree

at most

d ;

H will denote the maximum of the

heights

of a,

j3

and y.

After initial results of Ricci

[8]

and Franklin

[5],

Schneider

[9]

stated that for any E &#x3E;

0,

there exist

only finitely

many

triples (a, (3, y)

with

Bundschuh

[2]

remarked that in Schneider’s

proof

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;

0,

there are

only finitely

many

triples (a, (3, y)

with

where

log2

means

loglog. However,

there is an error at the

beginning

of the

proof

of his Satz

2a,

so that his

result,

too, is

only

valid if one

makes some extra

assumption.

Earlier

Smelev [10]

had

proved

that

only finitely

many

triples

(a, (3, y)

with

{3 ~ Q

have the

property

(3)

Cijsouw

and Waldschmidt

[4] recently improved

upon the above results

by showing

that for any

E &#x3E; 0,

there are

only finitely

many

triples (a, 03B2, y)

with

03B2 ~ Q

and

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

theorems,

the condition

{3 É 0

cannot be omitted. More

precisely,

the

following

will be

proved:

THEOREM 1: For any

fixed

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

(0, 1)

such that

for infinitely

many

triples (a, (3, 03B3) of

rational numbers

where H denotes the maximum

of

the

heights of

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

/3.

For

instance,

it is

quite

easy to see that the estimate of

Cijsouw

and Waldschmidt holds for

arbitrary triples (a, (3, y)

if one assumes

that,

for real

b,

the convergents

pnlqn

of the continued fraction

expansion

of b

satisfy

(Note

that the real numbers b for which this condition is not

fulfilled,

are U*-numbers

(see [9],

III

§3)

and thus form a set of

Lebesgue

measure

zero.)

A

sharper

result in the same direction is

given by

the

next theorem.

THEOREM 2:

Suppose E &#x3E; 0, dEN,

a,

bEC, a~0, b ~ Q ,

1 a

branch

of

the

logarithm

with

l(a) ~

0.

If b e R,

or

if

b ~ R such that

the convergents

Pnl qn of

the continued

fraction expansion of

b

satisfy

there are

only finitely

many

triples (a, (3, y) of algebraic

numbers

of

degree

at most d with

(4)

where H denotes the maximum

of

the

heights of

a,

i3

and y and

a b

= exp

(bl(a)).

In the

proofs

of Theorems 1 and

2,

the

following

notations will be

employed:

if a is an

algebraic number, [a[ (

denotes the maximum of the absolute values of the

conjugates

of a,

h(a)

the

height

of a,

dg(a)

the

degree

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

(0, 1)

such that

for

all n E N the

following inequalities

hold:

PROOF: Let

{31

be any rational number in

(0, 1)

with

h({31)

&#x3E; À. The sequence

({3n)~n=l

will be defined

inductively; suppose 6,, already

chosen.

Clearly

there are

infinitely

many rational

numbers 03B2*

E

(0, 1)

with the

property

Only finitely

many rational numbers have

heights

bounded

by

exp

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

so there exists a

03B2n+l with

both

h (03B2n+1)

&#x3E; exp

(h K+2(03B2n))

and

LEMMA 2: Let K, À EN be

given

and let

({3n)::’=l

be a sequence

of

rational numbers in

(0, 1)

such that

(1), (2)

and

(3)

are

satisfied for

all

n E N. Put

(3n

=

vn/wn,

where Vn, Wn E N,

(vn, wn)

= 1.

If

À is

s uffi cien tly large,

there is a sequence

(an)1lJ=1 of

rational numbers in

(0, 1),

such

that

for

all n E N the

following

assertions hold :

(5)

PROOF: Choose

al:= 2-wl.

The sequence

(an)~n=1

will be defined

inductively;

suppose a i, ..., an have

already

been chosen and possess the desired

properties. By

Bertrand’s Postulate

([6],

Theorem

418),

there is a

prime

number Un+l with

Notice

that,

if À is

sufficiently large,

Consider the

partition

of the interval

(0, 1).

Take t E

{0,...,

Un+l -

1}.

Then

therefore the width of the

partition

D does not exceed

wn+ilun+i. By (8)

the interval

{x E (0,1): lan - xl exp (-logK+lh(an))}

has a

length greater

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

{l, ...,

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

(0, 1)

such that

(1)-(3)

are satisfied.

Put On

=

vn/wn,

where vn, wn E N,

(vn, wn)

= 1. We may suppose that À is

sufliciently large

in the sense of

Lemma

2;

let

(an)n-=,

be a sequence of rational numbers in

(0, 1)

such

that

(4)-(7)

are satisfied.

Put an

=

(tnlun)Wn,

where tn, Un E

N, (tn, Un) =

1. Define yn : =

a’-;

we have

so

yn E

Q and

h(yn)

= u,,- u,,- =

h(an).

Therefore

II. The sequence

(an):=l

has the

property

if wi is

sufficiently large,

so x E

Ik-,.

III. From

(9)

we see that

(an):=l

is a

Cauchy

sequence; it converges

to a

limit,

which we shall call a. Then

(7)

which, by

Theorem 186 of

[6], implies

that a is irrational.

In the same way one can prove that

({3n);=1

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

compact

subset K of

(0, 1)

x

(0, 1),

so that a constant

CK, only depending

on

K,

can be found with

From this follows the existence of a constant

Ca,b, depending only

on

a and

b,

such that

for

sufficiently large

n. 0

It may be of interest to note that a

p -adic analogue

of Theorem 1

can be

proved

with

considerably

less

difficulty. Indeed,

it suffices to

construct a sequence

({3nr:=1

of natural numbers with the

properties /{3n - {3n+llp

exp

(_{3:+1)

and

/{3n/p

=

p-2.

If b is the

p-adic

limit of this sequence and

a = b + 1, infinitely

many

triples (a, (3, y)

of natural

numbers

satisfy

where H = max

(a, (3, y)

and

ab

is defined

by

means of the

p-adic logarithm

and

exponential

function.

4. A result on

vanishing

linear forms

LEMMA 3:

Suppose

d E tBI, K a compact subset

of

the

complex plane

not

containing 0, It

and

12

branches

of

the

logarithm, defined

on

K,

such that

11

does not take the value 0. Then

only finitely

many

pairs

(a, y)

E K X K

of algebraic

numbers

of degree

at most d have the

(8)

property that a

8

El 0 exists with

and

where H = max

(h (a ), h(y)).

PROOF: I.

Suppose

a, y E

K, (3

E

Q,

such that the conditions of the lemma are fulfilled.

By

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

depending only on d, K, Il

and

12;

we suppose that H is

greater

than such a

number,

which will lead to a contradiction.

P ut B : =

h(3);

then

Define

L: = [2dB log-1/3B] -1.

We introduce the

auxiliary

function

where aÀ1z = exp

(À izli(a )) , 03B303BB2z

= exp

(À2z12(’Y))

and where

p(À1, À2)

are rational

integers

to be determined later. We have

Now

put a : = den(a ), b:=den({3), c :=den(y), S:=[logl/3B],

T : =

[B21og-1B]

and consider the

system

of linear

equations

These are ST

equations

in the

(L

+

1)2

unknowns

p(Ài, À2);

the coeffi- cients are

algebraic integers

in the number field

Q(a, y)

of

degree

at

most d2. The absolute values of the

conjugates

of the coefficients are

less than or

equal to

(here (10)

is

used).

(9)

As

(L+1)2 ~ d2B2 log-2/3 B ~ d2ST,

Lemme 1.3.1 of

[11] ]

states that there is a non-trivial choice for the

p(Ài,À2),

such that

while

II. For kEN U

101

we

put Tk :

=

2kT;

suppose

2k ~ logl/6 B. Then,

for

our

special

choice of the

p(À1, À2),

we have

This is

proved by induction;

for k = 0 the assertion is

precisely (11).

Now suppose that

(12)

holds for

some k,

while

2k+l ~ logl/6B. By

Lemma 7 of

[3]

we have

Here

and

Substitution in

(13) gives

(10)

so

from which we conclude

However, l-;t(a )tP(t)(s)

is

algebraic

and formula

(1.2.3)

of

[11]

states that every non-zero

algebraic number e

has the

property

Now

so either

in the latter case

Combining (15)

and

(17) gives 0("(s)

= 0 for s =

0,...,

,S -

1,

t =

0, ..., T,,,, -

1. This

completes

the

proof

of

(12).

III. Now let k be the

largest

natural number with

2k ~ logl/6 B.

From

(12)

it follows that

Once more

apply

Lemma 7 of

[3];

this

gives (13) again

and

(14)

also

remains

unchanged,

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

Combining (18)

and

(19) gives

(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

type,

must be

singular.

From this we deduce the existence of

kl, À2, Ai, À2E {0,

...,

LI

with À +

À2{3

=

À1

+

À2{3,

or

This

gives

so we

get

a contradiction for

sufficiently large

H

(and B).

D

5. Proof of Theorem 2

1. The case

b ~

R is

trivial;

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

expansion

has the

property

described in the theorem. Let

(a, (3, y)

be a

triple fulfilling

the conditions of the

theorem;

we

suppose H

to be

greater

than a certain bound

depending only

on E,

d, a, b

and 1. This will lead to a contradiction.

As a ~ 0 and

a ~ 0,

we may assume a ~ 0 and

y ~ 0.

For

suitably

chosen branches

li

and

12

of the

logarithm

we have

from

l (a ) ~

0 we thus

get li(a) ~

0. As a consequence of

(21), (22)

and

we have

If it were the case that

(311(a) -12( ’Y) ~ 0,

Theorem 1 of

[4]

would

imply

which is a contradiction. Therefore

(311(a) -12( ’Y)

= 0.

II. We have

just proved

that

1,(a)

and

l2(y)

are

linearly dependent

(13)

over the field of all

algebraic numbers; using

Theorem 1 of

[1]

we find

that these numbers must also be

linearly dependent

over 0. In other

words,

there are

e, 17

E

Q,

not both zero, such that

03BEll(a)

+

~l2(y)

= 0.

Here q 0

0 because

I1(a) ~ 0,

so

using

Lemma 3 above we see that

h({3) log

H.

Put q :

=

den({3);

then q

log H,

so

As q must tend to

infinity

with

H,

we may assume

and

thus, by

Satz 2.11 of

[7], {3

is a

convergent of b,

say

{3

=

Pnl qn. By (12) in § 13

of

[7],

we

have,

for some constant c,

which contradicts

(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.

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[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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