Linear independence of continued fractions

J

AROSLAV

H

AN ˇ

CL

Linear independence of continued fractions

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{2 (2002),}

p. 489-495

<http://www.numdam.org/item?id=JTNB_2002__14_2_489_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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

Linear

_independence

of continued fractions

par JAROSLAV

HANCL

RÉSUMÉ. Nous donnons un critère

d’indépendance

linéaire sur le corps des rationnels

qui s’applique

à une famille donnée de

nom-bres réels dont les

_{développements}

en fractions continues satisfont

certaines conditions.

ABSTRACT. The main result of this paper is a criterion for linear

independence

of continued fractions over the rational numbers.

The

_proof

is based on their

special properties.

1. Introduction

Forty

years ago

Davenport

and Roth in

[2]

proved

that the continued fraction

_{[01,02?’" L}

where _{al, a2, ...} are

_positive

_{integers satisfying}

is a transcendental number. The

generalization

of transcendence is

alge-braic

_independence

and there are several results

_concerning

the

algebraic

independence

of continued fractions.

_See,

for

_instance,

Bundschuh

_[1]

or

Hancl

_[5].

On the other hand it is a well known fact that if a

_positive

real number has a finite continued fractional

expansion

then it is a rational

number,

and if not it is an irrational number.

_{Irrationality}

is a

special

case

of linear

_independence

and this _paper deals with such a

_{theory. By}

the

way, as to linear

_independence

of

_series,

one can find the criterion in

[4],

for instance.

2. Linear

_independence

Theorem 2.1. Let E &#x3E; 1 be a real

number,

K be a natural number and

=

1, 2, ... ,

K)

be K _sequences

of

positive integers

such that

and

Manuscrit reçu le 6 novembre 2000.

number 1 are

linearly independent

over the rational numbers.

Lemma 2.1. Let

_{1, 2, ... , K, n}

=

1, 2, ..

and K _&#x3E; 2

satisfy

all

conditions stated in Theorem 2.1. Then

Proof of

Lemma ,~.1. From

₍₁₎

and

₍₂₎

we obtain

for _every

_{sufficiently large}

_positive

_{integer n and j}

=

1, 2, ... ,

K.

By

math-ematical induction we

_get

for every n =

2, 3, ...

and j =

1, 2, ... , I~,

where Y is a

positive

real

constant which does not

_depend

on n. It follows that

because the series

is

_convergent.

_(To

_prove this last fact one can use Bertrand’s criterion for

convergent series,

for instance. See

_[3]

for

_example.)

0

Proof of

Theorem 2.1. If K =

1,

then

al has an infinite continued fraction

expansion.

In this case _al is irrational and Theorem 2.1 holds. Now we

will consider the case in which K &#x3E; 2 and n is a

sufficiently large positive

integer.

Let us assume that there exist K+ 1

integers A1,

A2, ... ,

AK,

We can write each continued fraction _aj

( j

=

1, 2, ... ,

K)

in the form

where

_q

=

aj,2, ... , is the n-th

partial

fraction of aj and

Rj,n

is the

remainder.

For

_Rj,n

we have the estimation

and

where c &#x3E; 0 is a constant which

depends only

on _{al, a2, ... , aK.}

(For

the

proof

see, for

instance,

[6].)

Substituting (4)

into

(3)

we obtain

Multiplying

both sides of the last

_equation

_by

we obtain

This

_implies

where

Mn

is an

integer.

First we will _prove that &#x3E; 0. Let P be the least

_{positive integer}

such that 0.

_(Such

a P must exist because not _every

_Aj

is

_equal

to

zero.)

Then we have

where B is a

_positive

real number and C is a constant which does not

depend

on n. We also have

for

_{every j}

=

1, 2, ... ,

K, n

=

1, 2, ...

which can be

proved by

mathematical

induction

_using

(This

identity

can be

found,

for

instance,

in

[6].)

(8)

and

(9)

imply

This,

Lemma 2.1 and

₍₁₎

_imply

where D &#x3E; 0 is a constant which does not

_depend

on n. From

(10)

and the

fact that we obtain

for _every

_{sufficiently large positive integer}

n.

Now we will _prove that

IMni

]

1 for n

sufficiently large.

From

(7)

we

obtain

This and

₍₅₎

imply

From this and

₍₁₎

we obtain

where F =

EK 11 Aj

I

is a

_positive

real constant which does not

_depend

on n.

(9)

and

(12)

_imply

From this and Lemma 2.1 we obtain

where H &#x3E; 0 is a constant which does not

_depend

on n.

(1),

(2)

and

(13)

where L is a

_positive

real constant which does not

_depend

on n. It follows

that

_IMnl

₍

1 for _every

_{sufficiently large}

_positive

_integer

n. This and

(11)

imply

that 0

_{I Mn}

_I

1 for _every

_sufficiently

_large

_n, where

_Mn

is an

integer.

This is

_impossible

therefore the numbers ai , a2, ... , aK and 1 are

linearly independent

over the rational numbers. 0

3. Conclusion

Example

1. The continued fractions

and the number 1 are

linearly independent

over the rational numbers.

Example

2. The continued fractions

and the number 1 are

linearly independent

over the rational numbers.

Example

3. The continued fractions

and the number 1 are

_{linearly independent}

over the rational numbers.

Open

Problem. It is not known if the continued fractions

and the number 1 are

linearly independent

or not over the rational numbers.

Example

4. Let be the linear recurrence _sequence of the

k-th order such k-that

_{G1, G2, ... , G~, bo, ... , b~ belong}

to

_{positive integers,}

G1

G2

...

_Gk

and for _every

_{positive integer}

_n,

_{Gn+k =}

_Gnbo

+

Gn+lb1

+ ~ - ~ + If the _{roots 0:1, ... as} of the

_equation

xk =

bo+b1x+...+bk-lXk-l

satisfy

jail

&#x3E;

_la2)

&#x3E; ... ~

_{la.1, 10:11&#x3E;}

1 and is not a root of

_unity

for

_{every j}

=

2, 3, ... ,

_s, then the continued fractions

( j

=

1, 2, ... ,

k)

and the number 1 are

linearly independent

over the rational

numbers.

This is an immediate _consequence of Theorem 2.1 and the

inequality

Gn

which can be found in

[7],

for instance.

Acknowledgments.

We would like to thank _{you very} much to Professor James Carter and Professor Atilla Peth6 for their

_help

with this article.

References

[1] P. BUNDSCHUH, Transcendental continued _fractions. J. Number Theory 18 _(1984), 91-98.

[2] H. DAVENPORT, K. F. ROTH, Rational approximations to _algebraic numbers. Mathematika 2 _(1955), 160-167.

[3] G. M. FICHTENGOLC, Lecture on _Differential and lntegrational Calculus II _(Russian).

Fiz-matgiz, 1963.

[4] J. _{HANCL, Linearly} unrelated sequences. Pacific J. Math. 190 _(1999), 299-310.

[5] J. HANCL, Continued _{fractional algebraic independence of} sequences. Publ. Math. Debrecen 46 _(1995), 27-31.

[6] G. H. _HARDY, E. M. _WRIGHT, An Introduction to the Theory of Numbers. Oxford Univ.

Press, 1985.

[7] H. P. SCHLICKEWEI, A. J. VAN DER _POORTEN, The _growth conditions for recurrence

se-quences. Macquarie University Math. Rep. 82-0041, North Ryde, Australia, 1982. Jaroslav HANCL Department of Mathematics University of Ostrava Dvotakova 7 701 03 Ostrava 1 Czech _Republic hancloosu. cz