Symmetry and folding of continued fractions

A

LFRED

J. V

AN DER

P

OORTEN

Symmetry and folding of continued fractions

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{2 (2002),}

p. 603-611

<http://www.numdam.org/item?id=JTNB_2002__14_2_603_0>

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

Symmetry

and

_folding

of continued

fractions

par ALFRED J. VAN DER POORTEN

To Michel France on his 65th birthday

RÉSUMÉ. Le ’lemme de

_pliage’

de Michel Mendès France

four-nit une nouvelle

justification

de la

_symétrie

du

développement

en

fraction continue d’un irrationnel

_quadratique.

ABSTRACT. Michel Mendès France’s

_’Folding

Lemma’ for

contin-ued fraction

_{expansions provides}

an unusual

explanation

for the

well known _symmetry in the

_expansion

of a

quadratic

irrational

integer.

1. Continued fractions

A continued

_frdction

_expansion

is

_{conveniently represented by}

_{[ago ,}

_{al ,}

_{a2 , ... .}

Set

_{[ ao ,}

al , ... ,

ah] =

Xh/Yh

for h =

_{0, 1, 2,}

.... Then the definition

( ao ,

al ,

... ,

ah

]

=

ao

+ 1 / [ al ,

... , ah

I

immediately implies by

induction on h that there is a

correspondence

between certain

_products

of two

_by

two matrices and continued fractions. It follows

_{by transposition}

of matrices that

Manuscrit _requ le 5 f6vrier 2001.

and

_by

_taking

determinants that

Proposition

2. One then sees that

It follows that if a :

Specifically,

if the

_partial

_quotients

_{ai, a2, ...} are

_{positive integers,}

as is

the case for the

simple

continued fraction

_expansions

we discuss

below,

and

a is

irrational,

then

illustrating

the

_quality

of

_{approximation}

of the

_convergents

_xh/yh

of a.

Continued fractions are well studied

_objects,

so new

_elementary

results

concerning

their structure are a

surprise.

Proposition

3

_(Folding

Lemma

_[2]).

We have

Here we have _{set Wh} =

al, a2 ... , ah;

accordingly

the notation

-w

de-notes -ah, -ah-17- -- , -al. It will save _space to write a for -a.

Proof.

Here E--~ denotes the

_{’correspondence’}

between two

_by

two matrices and continued fractions. We have

as

_alleged.

Moreover,

( c -

== c ,

]

by Proposition

1. 0

The

_adjective

_{’folding’}

is

_appropriate

because iteration of the

_perturbed

symmetry w -

w, c,

-iii

yields

a

_pattern

of

signs

on the word w

corre-sponding

to the

_pattern

of creases in a sheet of _paper

_repeatedly

folded in

half;

see

[1].

Incidentally,

it is often convenient to

_{drop subscripts}

and to use the

convention

_{whereby if,}

say, y denotes yh, then

y’

denotes the

previous

y;

that

_is,

_y‘

= _{yh-i .}

The

_folding

lemma makes it easy,

given

the

_expansion

of a

_partial

sum

of a

_series,

to

_adjust

a continued fraction

_expansion

for an

_appended

term

if the

_{intervening ’gap’}

is wide

_enough.

Our

_example

is from the function field _case, where

_’degree’

in X

_replaces

the

_logarithm

of the absolute value.

of

_{degree -2n}

or less. Of course if the

_degree

of the

_appended

term is much

~ less,

that introduces a

partial

quotient

of

correspondingly high degree.

As an

application

of the lemma: the continued fraction

expansion

of the sum

is

_given

_sequentially

_by

J

where the

addition of each term is done

_by

a ’fold’ with c =

-X ;

_see

~4~ * .

2.

_Symmetry

The

_computation

shows how to make

_{negative partial quotients}

_positive.

_Specifically

Here we use

having

recalled that

Thus

_folding

_yields

a

_symmetric

continued fraction

expansion, perturbed

(cf

[3])

in the

_{given example by}

the word _{ah ,} c -

1,1,

ah - 1.

This raises the

_question

as to whether _every such

_{’symmetric’}

_expansion

arises from

_folding.

Mind _{you, it} should not raise any such

thing.

If the

given

symmetric

expansion

yields

the

_convergent

_X/Y

and its ’first half’ is

x / y

then

_necessarily

is of the for some

_appropriate

c.

So,

a better

_question

is to ask what one

_might

learn from

_viewing

a

sym-metric

_expansion

as a folded

_expansion.

The best known case of a

sym-metric

_expansion

is the

_period

of a real

quadratic

irrational

integer.

Below

*

Jeffrey Shallit _recently noticed that the evaluation _{just given} is foreshadowed in remarks of Walter Leighton and W. T. Scott, ’A _general continued fraction expansion’, Bull. Amer. Math. Soc. 45 ₍₁₉₃₉₎ 596-605.

3. Periodic continued fractions

Suppose

the real

_quadratic

irrational

_integer

6 has norm n and trace t.

That

_{is, b2 -}

t6 -I- n = 0 with

_{integers n and t, whe_re t2 -}

4n _&#x3E; 0 is _{not a}

square in Z. To fix

6,

we _suppose that 6 &#x3E;

_6,

where 6 denotes the

_conjugate

of 6.

Let

_{ao ,}

al ,

... , ah ~ - ~ ao , ]

be an initial

_segment

of the

continued fraction

_expansion

of 6. We will

play

with the matrices

noticing

first that the

_{decomposition}

defines _sequences

_(Ph)h&#x3E;o

and

_(Qh)h&#x3E;o

of

_integers.

_Here,

the

determinant x 2 -

_txy

+

ny2

of

_{Nh .}

Also

Because

_x/y

is a

_convergent

of 6 we have

1/y,

and so

Q

is at

most 6 - 3. It follows from several

_applications

of the box

_principle

that there are

_positive

_integers

X and Y

_providing

a solution to ’Pell’s

_equation’

Proposition

4. tXY +

nY2

= ::1:1 and

X/Y

=

( ao ,

... , then

By

the matrix

_{correspondencet}

this is

Notice moreover that

Thus the word _{ao, al, ... ,} ar + t must be a

palindrome~.

t There is a subtle _point here at which I tacitly suppose that X 2 - tXY + ny2 = _{( -1 )’’ .}

However, that loses no _generality, because _{if X/Y} = _{[... ar also X/Y} = _[...

ar - 1 , 1 ). t Anubis: Dog as a devil deified lived as a _god.

4.

_Halfway

in the

_period

Suppose, first,

that the

_period

of 6 is of even

_length

r = 2h _{+ 2.} Then

N =

Nh

corresponds

to half the

period

in the sense that

N2

= (~

x Y

provides

X/Y

yielding

the

_period

as in

Proposition

4.

Thus we consider

whence

By

(2)

we have

and so,

indeed,

the

Folding

Lemma entails that

Proposition

5.

_Suppose

that the

_quadratic

irrational

_integer

6

_satisfies

6 &#x3E;

3

and has a continued

_fraction

_expansion

_{period of}

even

_length

2h + 2.

Then

if

and

_{only if Q}

divides 2P + t.

Proof.

The

_necessity

of the condition is clear since otherwise the

_given

expansion

is not admissible. For its

_sufficiency

we need to confirm that

X/Y

does

_yield

a solution to ’Pell’s

_equation’.

That is not obvious:

_plainly

X2 -

tXY +

ny2

=

Q2.

However,

it turns out that both

_QIX

and

_{QI Y.}

To see

_that,

consider

the conditions

₍₂₎

modulo

_Q,

that

_is,

Px - _-ny and

_{Py - x -}

_ty.

Recall we _suppose 2P+t - 0. Then X

= x2 -ny2 ==

= 0 and Y =

2xy-ty2

= =

(Py+(P+t)y)y -

_0,

as

_suggested.

0

Suppose,

second,

that the

_period

of 6 is of odd

_length

r = 2h ₊ 1. Then N

corresponds

to half the

_period

in the sense that

_provides

whence

By

(2)

and so

provided

that

_Q

=

Q’;

here w’ denotes the word

aI, a2,... , ah-1.

Proposition

6.

_Suppose

that the

_quadratic

irrational

_{integer J satisfies}

6 &#x3E;

3

and has a continued

f raction

_expansion

_{period of}

odd

length

2h + 1.

Then

if

and

_{only if Q}

=

Q’.

Remark. We will see below that

_Pk

+ + t =

akqk

for k =

_{0, 1,}

... , so

when

_Q’

=

Q

the

requirement

₊ P ₊

t)

is

automatically

satisfied.

Proof.

The

_necessity

of the condition is clear since otherwise the

_given

expansion

is not

_symmetric;

in

_particular

we would not have _ah =

(P’

₊

P +

_{t) /Q.}

For its

_sufficiency

we need to confirm that does

yield

a

solution to ’Pell’s

_equation’.

That’s not obvious:

_plainly

=

_QIQ = -Q2.

However,

both

_{Q I X}

and

_{Q I Y.}

Recall we have

Q’

=

Q

and _so P’ ₊

5. Remarks and

_explanations

5.1. Some identities. The

_equations

Px +

_{Qx’ =}

-ny and

Py

+

_{Qy’ =}

x -

_ty

_together

entail that

Brute

_computation

now shows that

We will avoid the need for such

_{brutality by showing}

that our also

taking

the

_equations

₍₂₎

in the form

_{P’x’ -f- Q’x" - -ny’}

and

_{P’y’ +Q’y"}

=

x’ - ty’

provides

a base from which one can _prove the facts we want

_by

induction. Here we note that x =

ax’

+ x"

and y

=

ay’ + y",

where _a = _ah. Then

and

Multiplying

these

_equations

_{respectively by x}

and

_by

_y, and

_subtracting,

yields

That

_is,

as

promised,

As

_just

_noted,

we therefore have

and

Multiplying

these

_equations

_{respectively by}

n and

_{by P,}

and

_{adding, yields}

is

_{step h}

in the continued fraction

_expansion

of J. Just so,

(4)

fixes the

reciprocal

of the remainder in

₍₅₎

above and thus

_Qh+l

and the next

com-plete quotient

(6

~-

_Moreover,

because the

_Q’s

are

_positive

it

follows from

₍₄₎

alone that -a

-3

for h =

0, 1,

.... Mind you,

we also know that the

complete quotients satisfy

(6

+

_P)/Q

&#x3E; 1 and the remainders -1

_{(~ + P)/Q’}

0.

Given the rules

₍₃₎

and

₍₄₎

_defining

the continued fraction

_expansion

it is an

_interesting

exercise to retrieve the conditions

₍₂₎

and so the

decompo-sitions

₍₁₎

of the matrices

_Nh.

The trick is to use

induction,

showing

that

+

_-ny’

and

_P’y’

+

_{Q’y" = x’ - ty’}

entails Px +

_{Qx’ =}

-ny

and

_Py

+

_Qy’

= _{x -}

_ty.

The

point

of this all is _to

emphasise

that the

two

_approaches

are

equivalent,

_confirming

that the matrices

Nh

detail the

continued fraction

_expansion

of 6.

5.3.

_Binary

_quadratic

forms. The matrix

_Nh

determines a

_quadratic

form

_(2P

+

_t)uv

+

(-1)hQ’v2

of discriminant

4p2

+ 4Pt +

t2

+

_4QQ’

= t2 -

4n

_{= (6 -}

_a)2.

_{That is, it provides}

P and

_Q,

and the

discriminant,

and therefore

_Q’.

It

_might

take us a little too far afield to

_{provide details,}

so I leave

con-firmation of the

_following

to the interested reader. It turns out that

mul-tiplication

of a

_pair

and

_{Nj-l is}

_composition

of the

_{corresponding}

quadratic

forms.

_Indeed,

this is a natural _way to define

_composition.

There

are

_provisos.

Whereas the forms determined

_by

the

_Nh

are

_reduced,

the

composite,

which a

priori

has

leading

coefficients is in

_general

not

reduced.

_Moreover,

it _may not be

_primitive.

Its coefficients will have the

common factor

G2,

where G =

gcd(Qt,

Q~,

+t).

One _must therefore

always

consider the

_{’primitive’}

matrix

_product

_G-1Ni-lNj-l.

5.4. The

_{’results’,}

_{specifically Propositions}

5 and

_6,

_given

here are of course well known. I doubt however whether

they

have heretofore been

proved by

the

_Folding

Lemma.

_My

remarks on the matrices

_Nh

contain

some

previously unpublished

observations. One motivation for

_viewing

continued fractions of

_quadratic

irrationals as a

_study

of the

_Nh

is that

doing

it

_suggests

an immediate

_{generalisation}

to

_{approximation}

_algorithms

for

_{higher degree algebraic}

irrationals. References

[1] F. M. DEKKING, M. MENDÈS FRANCE, A. J. VAN DER _POORTEN, FOLDS!. Math. _{Intelligencer}

4 _{(1982), 130-138;} II: Symmetry disturbed. ibid. 173-181; III: More _morphisms. ibid.

[2] M. MENDÈS FRANCE, Sur les fractions continues limitées. Acta Arith. 23 _(1973), 207-215. [3] M. MENDÈS FRANCE, Principe de la symétrie perturbée. Seminar on Number Theory, Paris

1979-80, 77-98, Progr. Math. 12, Birkhäuser, Boston, Mass., 1981. _{[MR 83a:10089]}

[4] A. J. VAN DER _POORTEN, J. SHALLIT, Folded continued fractions. J. Number Theory 40

(1992), 237-250.

Alf VAN DER POORTEN

ceNTR,e for Number _Theory Research

Macquarie University

Sydney, NSW 2109

Australia