Linear fractional transformations of continued fractions with bounded partial quotients

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J. C. L

AGARIAS

J. O. S

HALLIT

Linear fractional transformations of continued

fractions with bounded partial quotients

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{2 (1997),}

p. 267-279

<http://www.numdam.org/item?id=JTNB_1997__9_2_267_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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

Linear Fractional Transformations of Continued Fractions with Bounded Partial

_Quotients

par J.C. LAGARIAS ET J.O. SHALLIT

RÉSUMÉ. Soit 03B8 un nombre réel de développement en fraction continue

03B8 =

[03B10, 03B11, 03B12,... ],

et soit

une matrice d’entiers tel que det M ~ 0. Si 03B8 est à quotients partiels bornés, alors

_{[a*0, a*1, a*2,...]}

est aussi à _{quotients partiels} bornés. Plus _{précisément,} si _{aj ~} K pour tout j suflisamment _grand, alors

_{a*j ~}

| det(M)|(K

+

2)

pour tout j suffisamment _grand. Nous donnons aussi une

borne _plus faible _qui est valable pour tout

_a*j

avec j ~ 1. Les demonstrations utilisent la constante _{d’approximation diophantienne homogène}

_{L~(03B8) =}

lim sup q~~ (q~q03B8~)-1.

Nous montrons que

ABSTRACT. Let 03B8 be a real number with continued fraction _expansion

03B8 = [a0, a1, a2,...],

and let

be a matrix with _integer entries and nonzero determinant. If 03B8 has bounded partial quotients, then

_[a*0,

_a*1,

_{a*2, ...}

_]

also has bounded partial quotients. More _precisely, if

_{aj ~}

K for all _{sufficiently large j,} then

| det (M)|(K

+

2)

for all _{sufficiently large j.} We also give a weaker bound valid for all _{with j ~} 1. The _proofs use the _{homogeneous Diophantine}

approximation constant

_L~(03B8)

=

lim sup q~~(q~q03B8~)-1.

We show that

det(M)|L~(03B8).

1. INTRODUCTION.

Let 0 be a real number whose

_expansion

as a

_simple

continued fraction

is

and set

where we

_adopt

the convention that

K(0) =

+oo if 0 is rational. We _say that 0 has bounded

_partial

_quotients

if

_K(O)

is finite. We also set

with the convention that

_Koo

_{(8) =}

+oo if 0 is rational.

_Certainly

_Koo

_{(8) ~}

K(8),

and is finite if and

_only

if

_K(B)

is finite.

A _survey of results about real numbers with bounded

_{partial quotients}

is

_given

in

_[17].

The

_property

of

_having

bounded

_{partial quotients}

is

equiv-alent to 0

_being

a

badly approximable

number,

which is a number 0 such

that

in which

_{Ilxll I}

=

min(x -

x)

denotes the distance from x to the

nearest

_integer

and _q runs

_{through integers.}

This note _proves two

_quantitative

versions of the theorem that if B has bounded

_{partial quotients}

and M

b

is an

_integer

matrix with

J

det(M) #

0,

also has bounded

_{partial quotients.}

co+d

The first result bounds

_{K ( °’B+6 )}

in terms of

_Koo(8)

and

_{depends only}

cO+d

on

THEOREM 1.1. Let 0 have a bounded

partial

_quotients.

an

_integer

matrix with

_0,

then

The second result _upper bounds

_{K( de+b}

in terms

_{0 )}

and

_depends

_P

THEOREM 1.2. Let 9 have bounded

_{partial quotients. If}

M =

[a dJ

is

Lc

d.

an

_integer

matrix with

det(M) #

0,

then

The last term in

_(1.4)

can be bounded in terms of the

_{partial quotient}

ao of

9,

since

Theorem 1.2

_gives

no bound for the

_{partial quotient}

_ao

:=

I i

of

Chowla

_[3]

_proved

in 1931 that

_2ad(K(O)

_{+1)3 ,}

a result rather

weaker than Theorem 1.2.

We obtain Theorem 1.1 and Theorem 1.2 from

_stronger

bounds that relate

_{Diophantine approximation}

constants of 9 and which appear

below as Theorem 3.2 and Theorem

_4.1,

respectively.

Theorem 3.2 is a

simple

consequence of a result of Cusick and Mend6s France

[5]

concerning

the

_Lagrange

constant of 0

_(defined

in Section

_2).

The continued fraction of can be

_{directly computed}

from that for

8,

as was observed in 1894

by

Hurwitz

[9],

who _gave an

_explicit

formula for

the continued fraction of 20 in terms of that of 0. In 1912 Chitelet

_[2]

gave

an

algorithm

for

computing

the continued fraction of from that of

0,

and in 1947 Hall

_[7]

also _gave a method. Let

M (n, Z)

denote the set of n x n

integer

matrices.

_Raney

_[15]

_{gave for each M}

= a

b

E

_{M (2, Z)}

with

-c

d ’

det(M) =1=

0 an

explicit

finite automaton to

_compute

the additive continued

fraction of from the additive continued fraction of 0.

cO+d

In connection with the bound of Theorem

_1.1,

_Davenport

_[6]

observed that for each irrational 0 and

_prime

p there exists some

_integer

0 a _p

such that 0’ = 0

+ ~

has

_infinitely

_many

_partial

_quotients

_{an(B’) &#x3E;}

_p. Mend6s France

_[13]

then showed that there exists some 0’ = 0 + P

_having

p

the

_property

that a

_positive

_proportion

of the

partial quotients

of 0’ have

an (0’) 2::

P.

Some other related results appear in Mend6s France

[11,12].

Basic facts

on continued fractions _appear in

[1,8,10,18].

2. BADLY APPROXIMABLE NUMBERS

0 =

~ao,

al,...]

]

is determined

by

and for n &#x3E; 1

_by

the recursion

The n-th

_{complete quotient}

an of 0 is

The n-th

_convergent

v’ _{of 0 is}

qn

whose denominator is

_{given by}

the recursion q_ 1 -

0, qo - 1,

and

an+lqn + It is well known

(see

[8, §10.7])

that

Since _an+1 + 1 and qn, this

implies

that

for n &#x3E;

0.

We consider the

_following

_{Diophantine approximation}

constants. For an

irrational number 0 define its

_type

_L(O)

_by

and define the

_{horraogeneous Diophantine approximation}

constant or

We use the convention that if B is

rational,

then

L(0)

=

L~(B) _

_+oo.

(N.B.:

some authors

study

the

reciprocal

of what we have called the

La-grange

constant.)

The best

approximation

properties

of continued fraction

_convergents

_give

The set of values taken

_by

over all 0 is called the

_Lagrange

spec-trum

_[4].

It is well known that

_{L~ (B) &#x3E; J5}

for all 0. If 0 =

[ao,

_at,

a2, ... ~,

then another formula for

_Loo(6)

is

see

[4,p. 1].

There are

simple

relations between these

quantities

and the

partial

quo-tient bounds

_K(O)

and cf.

_[16,

_pp.

_22-23].

LEMMA 2.1. For _any irrational 0 with bounded

_{partial quotients,}

we have

Proof.

This is immediate from

_(2.2)

and

_(2.3).

0

LEMMA 2.2. For any irrational 0 with bounded

partial

quotients

Proof.

This is immediate from

_(2.2)

and

_(2.4).

0

Although

we do not use it in the

sequel,

we note that both

inequalities

Since from some

_point

_on, this and

_(2.4)

_yield

Next,

from

_(2.1)

we have

Hence

Let K =

K,,. (0).

Then for all n

sufficiently large

we have

so

We conclude that

3. LAGRANGE CONSTANTS AND PROOF OF THEOREM 1.1.

An

_integer

matrix M

= La

d

_I

with

_{det(M) #}

0,

acts as a linear

[c

oj

fractional transformation on a real number 8

by

LEMMA 3.1.

_If

M is an

_integers

matrix with

det(M) =

_~1,

then the

La-grange constants

_of

0 and

_M(O)

are related

by

Proof.

This is

_well-known,

cf.

_[14]

and

_[5,

Lemma

_1],

and is deducible from

(2.5).

D

The main result of Cusick and Mend6s France

_[5]

_yields:

THEOREM 3.2. For _any

_integers

m &#x3E;

_1,

let

Then

_for

_any irrational number

_0,

and

Proof.

Theorem 1 of

_[5]

states that

Let

_{GL(2, Z)}

denote the _group of 2 x 2

_integer

matrices with determinant

~1. We need

_only

observe that for any M in

Gm

there exists some M E

GL(2, Z)

such that 1Vl M =

0 d’

with a‘d’ = m and 0 b’ d’ . For if

0

U

_d

so,

and 0 =

then Lemma 3.1

gives

whence

_(3.4)

_implies

_(3.2).

To construct M

Take C = and Then

_{gcd(C, D) -}

_1,

so we _may

complete

this row to a matrix

M

E

_{GL(2, Z).}

_Multiplying

this

_by

a suitable

matrix

c

yields

the desired

M.

1

0

_{±l ’}

y

The lower bound

_(3.3)

follows from the _upper bound

_(3.2).

We use the

adjoint

matrix

-We _prove

_by

contradiction.

_Suppose

_(3.3)

were

_false,

so that for some

M E

_Gm

and some 0 we have

This states that

which contradicts

_(3.2)

for

_0’,

since

_det(M’)

=

det(M)

= m. 0

Remark. The lower bound

_(3.3)

holds with

_equality

for some values of 0

and not for other values. If for

_{given 0}

we choose an M E

_Gm

which

_gives

equality

in

_(3.2),

so that = then

equality

holds in

(3.3)

for B’ =

adj(M)(0).

_However,

if =

V5,

_{as occurs} for 0 = then

for all

_M;

hence

_(3.3)

does not hold with

_equality

when m &#x3E; 2.

Proof of

Theorem 1.1. Theorem 3.2

_gives

:5

Now

apply

Lemma 2.2 twice to

_get

To obtain the lower

_bound,

we use the

_adjoint

J

Since

_{I det (M) I =}

this

_yields

4. NUMBERS OF BOUNDED TYPE AND PROOF OF THEOREM 1.2

Recall that the

_type

_L(8)

of 0 is the smallest real number such that

THEOREM 4.1. Let 0 have bounded

_{partial quotients.}

an

_integer

matrix with

_{det(M) #}

_0,

then

Proof. Set 1b =

Suppose

first that c = 0 _so that

I det(M)1 = ladl

_&#x3E; 0.

Then

L(0)

&#x3E;

_1,

where

We have

For any e &#x3E; 0 we _may choose _q in

(4.2)

so

that ~ ~

L(1f;) -

e. Then

Letting

c -~ 0

_yields

_(4.1)

when c = 0.

We have

so that

We first treat the case _{qa - pc} = 0. Now

since det =

det(M) #

0. Thus if qa - pc = 0 then

lPd -

_1,

.J

hence

_(4.5)

_gives

It follows that _{qa - pc}

_,-~

0

_provided

that

~/

We next treat the case

_{when qa -}

_pc

₅₄

_{0. Now from the definition of} we see

Given e &#x3E;

_0,

we _may

_{choose q}

so

that ~ ~

_{L(~) -}

_{e, and} we obtain _from

(4.6)

and

_(4.9)

that

implies

that

Multiplying

this

_{by §}

and

_applying

it to the left side of

_(4.10)

yields

q

Letting

e -7 0 and

_{using q &#x3E;}

1

_yields

provided

that

_(4.8)

holds. Now

_(4.8)

fails to hold

_only

if

The last two

_{inequalities imply}

_(4.1)

when 0. 0

Proof of

Theorem

_Applying

Theorem 4.1 and Lemma 2.1

_gives

which is the desired bound. 0

Remarks.

_(1).

The

_proof

method of Theorem 4.1 can also be used to

directly

prove the bounds

of Theorem

_3.2,

from which Theorem 1.1 can be

easily

deduced. The lower

bound in

_(4.14)

follows from the _upper bound as in the

proof

of Theo-rem 3.2. We sketch a

proof

of the _upper bound in

(4.14)

for the case

9 =

with 0. For _any c* &#x3E; 0 and all

_{sufficiently large}

_{q* &#x3E; q*}

_{(E* ),}

we have

We

_{choose q}

= _qn

(1b)

for

sufficiently

large

_n, and note that

as n -~ _oo,

_{since 0}

is irrational. We can then

replace

(4.9)

by

(4.15),

and

then deduce

_(4.11)

with

_L(O)

_{replaced by}

+,E*.

_Letting

q -+ _oo,

f -+ 0 and E* -~ 0 in that order

_yields

the _upper bound in

_(4.14).

(2).

For a

_given

matrix M consider the set of attainable ratios

By

Lemma 3.1 the set

_V(M)

_{depends only}

on its

SL(2, Z)-double

coset

Theorem 3.2 shows that

It is an

_interesting

_open

_problem

to determine the set Both

_I

and

I lie in

V(M),

as follows from Theorem 3.2 and the remark

following

it.

Acknowledgment.

We are indebted to the referee for

helpful

comments and

_references,

and in

_particular

for

_raising

the open

problem

about

V(M) .

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J. C. _Lagarias

AT &#x26;T Labs - _Research, Room C235 180 Park _Avenue, P. (J. Box 971 Florham _Park, NJ _07932-0971, USA email: _{jcl@research.att.com}

J. 0. Shallit

Department of Computer Science University of Waterloo

Waterloo, Ontario N2L _3G1, Canada email: shallit@uwaterloo.ca