On a theorem of Legendre in the theory of continued fractions

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

D

OMINIQUE

B

ARBOLOSI

H

ENDRIK

J

AGER

On a theorem of Legendre in the theory of

continued fractions

Journal de Théorie des Nombres de Bordeaux, tome

6, n

o

_{1 (1994),}

p. 81-94

<http://www.numdam.org/item?id=JTNB_1994__6_1_81_0>

© Université Bordeaux 1, 1994, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

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

On

a

Theorem

of

Legendre

in

the

_theory

of continued fractions

by

DOMINIQUE

BARBOLOSI and HENDRIK JAGER

1. Introduction

Let r be a rational

number,

r = with

A,

B _E

Z,

B ~

0. We shall

always

assume that

(A, B)

= 1 and that B _&#x3E; 0. A rational number

A/ B

has two

_{representations}

as a continued fraction:

- if

B ~

1,

then

and

If the continued fraction

expansion

of

_A/.B

is determined

_by

the

Eu-clidean

algorithm,

the outcome is the

_expansion

_{( 1.1 ),}

i.e. the shortest one.

In this note we consider the shortest

_expansion

as the most natural one, we

call it the

_regular

continued fraction

_expansion

of

_A/B.

The

_regular

con-tinued

fraction

_expansion

of an irrational

_{number ~}

is infinite and

_unique.

We shall denote it

_by

Let

be the sequence of

corresponding

convergents,

also denoted

by

RCF(~).

Hence E

_RCF(~)

means that there exists an

_{integer n, n &#x3E;-}

_-1,

such

that A 1 -1

(1.3)

DEFINITION. The

_{appro2zrnation}

_{coefficients of}

a rational number

A/B

a4th

respect

to a real irrational

number ~,

notation

0(~,

A/B),

is

de-fined by

- - - --- m - m - - m

In his "Essai sur la théorie des

nombres",

[13],

_pp.

_27-29,

Legendre

gives

a _necessary and sufficient condition for a rational number

A/B

to

be a _convergent of the irrational number

C.

This necessary and sufficient

condition is

_expressed,

in modern

_{notation, by}

_(2.4)

and

_(2.6)

of the next

section of this note.

_Legendre

concludes the

_paragraph

on the criterion

by

remarking

that in

_particular

it follows that:

In the

_theory

of continued

_fractions,

_(1.4)

is often called

_Legendre’s

Theo-rem.

The

_{following implication}

is almost trivial

The

_{constants -1}

and 1 are both best

possible.

For the

constant 2

in

_(1.4)

this means that for _{every e} &#x3E; 0 there exist

_{a ~}

and

_{an A}

such that

_0(~,

_~)

2 + E

and -A 0

RCF(~).

Similarly

for the 1 in

_~1.5).

In this note we shall add some refinements to

_Legendre’s

_reasoning

and thus find a more detailed version of

(1.4).

We shall also

show

that one

can _{prove in} the same _way a result announced

_by

Fatou in

_1904,

as well

as the

_analogues

of

Legendre’s

Theorem for two other

_types

of continued

2.

_Legendre’s

criterion

We shall from now on assume

that ~

and

A/B

are contained in the unit

interval,

this

_being

no restriction. Hence the _ao in

(1.1)

and the

ao(ç)

in

(1.2)

are both zero. Let n E N and

_(ai,

a2,... ,

an) E

Nn . Then we denote

the set of irrational

numbers C

with the

_property

that

by

Such a set is called a

_fundamental

interval

_of

order _n, see

[2]

_p. 42.

(2.1 )

DEFINITIONS. The

_signature

_e(A/B)

_of

a rational nurriber

_A/B,

is

defined

as

A A

urith the n taken

_from

the

regular

_expansion

(1.1).

This n is sometimes called the

depth

of

the rational numbers

_A/B.

F’urther we

define

Finally,

the

_signature

_of

a mtional number

A/B

with

_respect

to an irrational

notatiort

_{b(~, A/B),}

is

_{defined by}

Hence

_{b(~, A/B)}

is determined

_by

the

_depth

of

_A/B

and the order

_{of ~}

and

A/B.

We shall now formulate a more detailed version of

_Legendre’s

Theorem

_(1.4),

in which a distinction is made between

6(C,

A/B) _

+1 and

b(C, AIB)

= -1.

(2.2)

THEOREM. Let

_AlE

be a rational

_number,

_(A,

B)

=

_1,

B _&#x3E; 0 and

and

If

on the other hand

6(~,~) =

_-1,

_then

and

All constants are best

_possible.

(2.3)

Proof

Let the

_regular

_expansion

of

_A/B

be

_given

_by

_(1.1)

and _suppose that n

is even. Denote

_{by A’/B’}

_{the last but one}

_convergent

_of

_(1.1).

_{The set of}

all ~ with ~

&#x3E;

_A/B,

i.e. with

_6(ç,AIB)

=

_1,

and with

AIB

E

_RCF(ç)

is

just

the fundamental

interval.

of order _n, i.e. the set

Hence

Now

and thus

Since

the assertions are now evident.

Let ~

_A

(and n

still be

_even).

Then the set of

_{all ~}

_{with A}

E

_RCF(C)

is the fundamental interval of order n + 1 :

- ....

which has

length

B-1(2B -

B’)-l.

Instead of

_(2.4)

we now have

and the assertions follow in this _case,

_using

_again

_(2.5),

from

- . ’1 ’1 -

-respectively.

The

_proof

for the case where n is odd is almost the same; therefore we omit it here. We see that the constant

_1/2

in

_Legendre’s

Theorem is due to

ra-tional numbers

_A/B

with

_{b(~, A/B) _ -1}

and with a _very

_large

last

partial

quotient

an -

+

3. A metrical observation

(3.1 )

DEFINITION. The sequence

of

regular

continued

_fraction

approxima-tion

_{coefficients of}

a real irrational

number ~

is

defined by

For almost

all-in

the sense of

Lebesgue,

the _sequence is

dis-tributed in the unit interval

_according

to the function

_F,

where

( 1 - 1

see

[3].

The

irregular

behaviour of F at A =

1/2

_can be

explained

by

the

constant

_1/2

in

_Legendre’s

Theorem

_(1.4),

see

[6].

In view of Theorem

(2.2)

one _may ask

_why

F does not have an

irregularity

at A =

2/3.

The

answer is

_given

by

the next theorem which shows that there are in fact two

(3.3)

THEOREM. For almost

_{all ~}

one has

with

and

with

(3.4)

Proof

From the

_alternating

way in which the sequence converges to

~

and from the fact that in the definition of

_e(~,

_{9. (t) )}

the n is taken from the shortest

_expansion

of

_~~

as a continued

fraction,

it follows that

and that

After this remark the

_proof

can be

_given

_using

the same

_techniques

with

4. Extreme mediants and the Theorems of Fatou-Grace and Koksma

DEFINITION. Ttae _sequence

_of

mediants

_of

a real irrational

number ~

is the

sequences

of

irreducible

_{fractions of}

the

_from

ordered in such a _way that the denominators

form

an

ascending

_sequence,

compare

f9J

p. 26. The

fractions

in

_(4.~)

_formed

with b = 1 _are called the

first

mediants

of ~,

those with b =

o~-i(~) 2013 1

the last mediants

of £.

The

first

and the last mediants are called extreme or nearest mediants. A

_first

mediant is also a last one

if

and

only if

the

corresponding partial

_quotient

equals

2.

In

_1904,

P. Fatou stated that if

_6(~,A/J5)

1,

then

_A/B

is either a

convergent or an extreme mediant

_{of ~,}

_[4].

The first one to

_publish

a

_proof

of this was J. H. Grace

[5],

see also Koksma

[10]

and

[11].

We will therefore

,refer

to this result as the

_Theorem

of Fatou-Grace. Koksma

[11]

showed

that

_{B(~, A/B)}

_2/3

implies

that is either a

_convergent

or a first

mediant

of ~.

We will now _prove more detailed versions of these results

_by

the method from section 2.

(4.3)

THEOREM. Let be a rational

(A, B)

=

1,

B _&#x3E; 0 and

let ~

be an irrational number.

-

~)

=1,

then B is

not

_{a first}

mediant

_off..

8(~,

_B)

1 a

_convergent

or a

first

mediant

_ofç,

and

B(~,

B )

&#x3E; 2 ~ neither a

convergerit

nor a

_first

mediant

Both constants 1 and 2 are best

_possible.

Theorems

_(2.2)

and

_(4.3)

_yield

at once the

following

result of Koksma

([10]

p.

102):

(4.4)

THEOREM

_(KoxsntA).

_If

_A/B

is a

_rational,

_{and ~}

an irrational

number

and

_{if 8(f., A/B)}

_2/3,

then

_A/B

is either a

_convergent

or a

first

(4.5)

Remark. The constant

_2/3

is due to the constant

_2/3

in Theorem

(2.2)

i.e. from rational numbers

_A/B

with

_{b(~, A/B)}

= 1 and with last

partial quotient

2.

(4.6)

Proof of

Theorem

_(4.3)

Let

_A/B

have the

_expansion

_(1.1)

and suppose that n is even. The set

of irrational

_{numbers C}

such that

_A/B

is of the form

_{p‘‘t +p’‘_’}

for some

’+’qk-1

k,

is the fundamental interval

_{An+ I (a,,}

a2, , c -

1,1).

Hence,

when

_{b(C, A/B)}

=

_1,

A /B

_{is not a} first mediant

_{of C,}

whereas when

~(~,~4/B) = 20131,

the set of

~’s

such that

_A/B

is either a _convergent or a

first mediant

_{of C}

is

_just

the fundamental interval

which has

length

B’)-1.

Therefore,

if

_6(~,

_{A/B) _ -1,}

_A/B

is a

convergent

or a first mediant if and

_only

if

Using

(2.5)

we then find that

F

is a

_convergent

or a first mediant

of ~

is neither a _convergent nor a first mediant of

_~.

The statements now follow from

The

_proof

for the case where n is odd is almost the same.

+

(4.7)

THEOREM. Let

_A/B

be a rational

_number,

(A, B)

=

1,

B _&#x3E; 0 and

let ~

be an irrational number.

is a

_convergent

or a last mediants

_{of ~’,}

A

is a last mediants

of

is a

first

mediant

_{of ç,}

is a

_convergent

or a last mediants

of ~,

9(~, B)

&#x3E; 1 is neither a

_convergent

nor a last mediants

of ~.

All constants are best

possible.

(4.8)

Proof

’

The set of all irrational

numbers £

such that

_A/B

is a last mediant

of C

consists of the union of the two fundamental intervals

i.e. the

_ç’s

in the first interval of

_(4.9)

are those for which

A/B

is a first

and a last mediant. After these remarks the

proof

runs almost the same as

the

_proofs

of Theorems

_(2.2)

and

_(4.3)

and may therefore be

omitted. +

The location of the various intervals

_occurring

in this section and in section

2,

is

_depicted,

for even _{n, in} the

_figure

below. For odd _n, the order is

reversed.

5.

_Legendre’s

Theorem for two other

_types

of continued fractions The above method can be used to obtain similar results for other types

of continued fraction

_expansions.

We will illustrate this with two

_examples:

(1)

the continued fractions with odd

_partial

_quotients,

(5.1)

The continued fraction

_expansion

with odd

_partial

_quotients:

_every irrational

_{number ~}

in the unit interval has a

_unique

_expansion

of the form

with

(5.3)

bn

an odd

_{positive integer, s}

We will denote the _sequence of _convergents associated with the

_expansion

(5.2)

by

OCF(~).

A rational number

_A/B

_always

has a finite

_expansion

with the same conditions as in

(5.3).

If in

(5.4)

one has

bn

= 1

-1,

then admits two

_expansions

of this

_type,

viz.

and

otherwise such an

_expansion

of a rational number is

_unique.

We consider

the

_expansion

_(5.5)

as the most natural one since it is obtained when one

repeatedly applies

the shift

_operator

for this continued

_fraction,

_just

as in the case of

(1.1).

For a

_description

of this

_operator

the reader is referred

to

_[14].

(5.7)

THEOREM. Let

_A/B

be a rational

numbers,

(A, B)

=

_1,

B _&#x3E; 0 and

let ~

be an irrational number.

Here g

_{:= ~(ý’5 + 1),}

hence g2

= 0,3819... ,

and G := g-1

=

1, 6180 ~ ~ - ;

the

_four

constants are best

_possible.

It was

_already

known that

B(~, A/B)

&#x3E; G

_implies

_OCF(~),

i.e. the

_analogue

of

_(1.5).

We now also have the

_analogue

of

(1.4),

that is

Legendre’s

Theorem for the continued fractions with odd

_partial

_quotients:

(5.8)

COROLLARY. Let

_A/B

be a rational

number,

(A, B)

=

_1,

B _&#x3E; 0 and

let ~

be an irrational

If

then

_{AIB is}

a

_convergent

_of

the

_expansion

_of

_{into its continued}

_fraction

with odd

partial

quotients.

The constant

_{0, 3819 ...}

is best

_possible.

(5.9)

Proof

of

Theorem

_(5-7)

The

_proof

differs

only

in technical details from that of Theorem

_(2.2).

Therefore it may suffice to indicate the main differences. First note that

for

the

_E(AIB)

from definition

_(2.1)

one has

A

where n and _{el, E2, ~ ~ ~ ,} are

_given

by

the

_expansion

(5.4)

_and,

in case we have the two

_expansions

_(5.5)

and

_(5.6),

_by

_(5.5).

This can

easily

be

shown with the

_techniques

II 7.

Next,

denote

_by

_I(A/B)

the set of irrational

_{numbers ~}

such that

_A/B

E

OCF(~).

If

_A/B

has the

_expansion

_(5.4)

with

_{bn &#x3E; 3,}

the end

_points

of

I(A/B)

are

and

from which it follows that

and with the end

_points

reversed when

_{e(A/ B) = -1.}

Here

_A’/B’

denotes the last but one

_convergent

of

_(5.4).

If has the

_expansion

_(5.5),

then the end

_points

are

and

from which it follows that

with

_A’/B’

the last but one _convergent of

_(5.5)

and

_again

with the end

points

reversed when

_{~(~4/J3) = 20131.}

The

_analogue

of

_(2.5)

is

which follows from the structure of the two-dimensional

_ergodic

_system

underlying

this continued

_fraction,

see

[1]

and

[14].

The rest of the

_proof

is

exactly

the same as the

_{corresponding}

_part

of the

_proof

of Theorem

(2.2). +

(5.10)

Remark. The constant

g2

from

_Corollary

_(5.8)

_corresponds

to an

irregularity

of the distribution

_function,

for almost all

_~,

of the sequence of

approximation

coefficients connected with this continued

fraction,

see

[1J,

IV,

Théorème 1. One could

_give

here results similar to those in section 3.

Recently,

the

_analogue

of

_Legendre’s

Theorem for the nearest

_integer

con-tinued fraction

_expansion

was found in three different _ways, see

[7], [8]

and

[12].

The constant turns out to be the same as in the case of the

contin-ued fraction with odd

_partial

_quotients:

_g2.

The method from the

_previous

sections

_applies

to the nearest

_integer

continued fraction as well. The

se-quence of

convergents

of this

_expansion

is denoted here

_by

_NICF(C).

We will state the result without

_giving

the

_proof,

which is very similar to the

previous

ones.

(5.11)

THEOREM. Let

_A/B

be a rational

_numbers,

_{(A, B)}

=

1,

B _&#x3E; 0 and

let £

be an irrational number.

and

and

All constants are best

possible.

REFERENCES

[1] D. BARBOLOSI, Fractions continues à _{quotients partiels impairs, Thèse,} Univer-sité de Provence, Marseille

_(1988).

[2]

P. BILLINGSLEY, Ergodic Theory and _Information, John Wiley and Sons, New

York, London, Sydney (1965).

[3]

W. BOSMA, H. JAGER and F. WIEDIJK, Some metrical observations on the

approximation by continued fractions, Indag. Math. 4 _(1983), 281-299.

[4]

P. FATOU, Sur _{l’approximation} des incommensurables et les séries

trigonométri-ques, C. R. Acad. Sci. Paris 139

(1904),

1019-1021.

[5] J. H. GRACE, The classification of rational approximations, Proc. London Math. Soc. 17

_(1918),

247-258.

[6]

S. ITO and H. _NAKADA, On natural extensions _{of transformations} related to

Diophantine approximations, Proceedings of the Conference on Number Theory

and Combinatorics, Japan 1984, World Scientific Publ. _{Co., Singapore}

_(1985),

185-207.

[7]

S. _ITO, On _Legendre’s Theorem related to _{Diophantine approximations,} Séminaire de Théorie des Nombres, Bordeaux, exposé 44 _(1987-1988), 44-01-44-19.

[8]

H. JAGER and C. _KRAAIKAMP, On the _{approximation by} continued fractions, Indag. Math. 51 _(1989), 289-307.

[9]

J. F. _{KOKSMA, Diophantische Approximationen,} Julius _Springer, Berlin

_(1936).

[10]

J. F. _{KOKSMA, Bewijs} van een _stelling over _{kettingbreuken,} Mathematica A 6

(1937),226-231.

[11]

J. F. _KOKSMA, On continued _fractions, Simon Stevin 29 _(1951/52), 96-102.

[12]

C. _KRAAIKAMP, A new class _of continued _fractions, Acta Arith. 57 _(1991), 1-39.

[13]

A. M. _LEGENDRE, Essai sur la théorie des nombres, Duprat, Paris, An VI

_(1798).

[14]

F. _SCHWEIGER, On the _{approximation by} continued _fractions with odd and even

partial quotients, Mathematisches Institut der Universität _Salzburg, Arbeitsbericht 1-2

_(1984),

105-114.

Dominique Barbolosi

Universite de Provence, UFR de _{Math6matiques} 3 _place Victor _Hugo

F-13331 Marseille Cedex 3 France Hendrik _Jager Raadhuislaan 49 2131 BH _Hoofddorp Pays-Bas