Pages from JTNB 2002 14 2 497 0 decrypted

Algebraic

and

_ergodic

_properties

of

a new

continued fraction

_algorithm

with

non-decreasing partial quotients

par YUSUF HARTONO,

COR KRAAIKAMP

et FRITZ SCHWEIGER

RÉSUMÉ. On introduit la notion de

_{développement}

en fractions

continues de

_Engel.

Nous étudions notamment les

_propriétés

er-godiques

de ce

développement

et le lien avec celui introduit _par

F.

_Ryde

_monotonen, nicht-abnehmenden Kettenbruch.

ABSTRACT. In this paper the

Engel

continued fraction

(ECF)

ex-pansion

of any x ~

(0,1)

is introduced. Basic and

_ergodic

proper-ties of this

_expansion

are studied. Also the relation between the

ECF and F.

_Ryde’s

_monotonen, nicht-abnehmenden Kettenbruch

(MNK)

is studied.

1. Introduction

Over the last 15 years the

ergodic

properties

of several continued fraction

expansions

have been studied for which the

_{underlying dynamical}

_system

is

_ergodic,

but for which no finite invariant measure

equivalent

to

Lebesgue

measure exists.

Examples

of such continued fraction

expansions

are the

’backward’ continued fraction

_{(see ~AF~ ),}

the ’continued fraction with even

partial quotients’

(see

[S3])

and the

_Farey-shift

_{(see [Leh]).}

All these

_(and

other)

continued fraction

_expansions

are

ergodic,

and have a

a-finite,

infi-nite invariant measure. Since these continued fractions are

closely

related

to the

_regular

continued fraction

_(RCF)

_expansion,

their

_ergodic

proper-ties follow from those of the RCF

_{by using}

standard

_techniques

in

_ergodic

theory

(see

[DK]).

In this _paper we introduce a new continued fraction

expansion,

which we

call-for reasons which will become

apparent

shortly-the Engel

continued

fraction

_(ECF)

expansion.

We will show that this ECF has an

underlying

dynamical

system

which is

_ergodic,

but that no finite invariant measure

equivalent

to

_Lebesgue

measure exists for the ECF.

As the name

_suggests,

the ECF is a

generalization

of the classical

Engel

Series

_expansion,

which is

_generated

_by

the _map S’ :

_{[0, 1)}

-+

[0, 1),

given

by

I ’B.

where

_{[g J}

is the

_{largest integer}

not

_{exceeding ~,}

see also

Figure

1.

FIGURE 1.

_Engel

Series _map S

Using S,

one can find a

(unique)

series

expansion

of _{every x} E

(o,1 ) ,

given by

1 1 1

where qn

= +

1,

n > 1. In fact it _was W.

Sierpinski

[Si]

in 1911 who first studied these series

_expansions.

The metric

_properties

of the

_Engel

Series

_expansion

have been studied in a series of _papers

by

E.

Borel

[B],

P.

Levy

[L],

P.

Erd6s,

A.

R6nyi

and

P. Sz3sz

_[ERS]

and

_R6nyi

_[R].

In

_[ERS]

it is shown that the random vari-ables

_Xl

=

logql,

Xn

=

log(qn/qn-i)

_are ’almost

independent’

and ‘almost

identically

distributed’. From this first the central limit theorem is derived for

_{log qn,}

then the

_strong

law of

_large

numbers and

_finally

the law of the iterated

_logarithm

are obtained. The second result has been announced

earlier without

_proof

_by

E. Borel

_[B].

The first and third results are due

to P.

_Levy

_[L].

In

_[R],

_R6nyi

finds new

(and

more

elegant)

proofs

to these and other results. Later F.

_Schweiger

_[Sl]

showed that S is

_ergodic,

and M. Thaler

_[T]

found a whole

family

of

Q-finite,

infinite measures for S.

Further information on the

Engel

Series

(and

the related

Sylvester

Series)

can be found in the books

by

J. Galambos

[G]

and

Schweiger

[S3].

Clearly

one can

generalize

the

Engel

Series

expansion

by changing

the

time-zero

_partition,

or

by ’flipping’

the _map S on each

partition

element

Let be a

monotonically

decreasing

_sequence of numbers in

(o,1),

with _{rl =}

_1,

rn > 0 for n > 1 and 0.

_Furthermore,

let

en

=

1,

and let the time-zero

partition P

be

given by

Then

where n E N is such that x E

[fn)

rn)

_generalizes

the

_{’Engel-map’}

S. With

some effort the ideas from the above mentioned _papers can be carried over

to this

_{generalization,}

also see W. Vervaat’s thesis

(V).

In this paper we

study

a different variation of the

Engel

Series

expansion.

Let the

_Engel

continued

_fraction

_(ECF)

_map

_{TE :}

_{[0, 1)}

-

[0, 1)

be

given

by

see also

Figure

2. Notice that

where T :

_{[0, 1]}

~

[0, 1)

is the

regular

continued fraction _map,

given by

and = For _{any x} E

(0,1),

the

ECF-map ’generates’

(in

_{a way}

similar to the way the

RCF-map

T

’generates’

the

RCF-expansion

of

x)

a new continued fraction

expansion

of x of the form

In this _paper we will

study

in Section 3 the

ergodic properties

of this new

continued fraction

_expansion,

the so-called

_Engel

continued

_fraction

_(ECF)

expansion. However,

since the ECF is new, we will first show in the next

section that the ECF ’behaves’ in _{many ways} like any other

(semi-regular)

continued fraction. In the last section we will

study

the relation between

the

_ECF,

and a continued fraction

expansion

introduced

by

F.

Ryde

[Ryl]

in

_1951,

the so-called

_monotonen,

nicht-abnehmenden Kettenbruch

_(MNK).

We will show that the ECF and a minor modification of

Ryde’s

MNK are

metrically isomorphic,

and due to this _many

_properties

of the ECF-such

as

ergodicity,

the existence of

a-finite,

infinite measures-can be carried over to the MNK.

Conversely,

the fact that not _every

quadratic

irrational x

has an

ultimately periodic MNK-expansion

can be carried over to the ECF via our

isomorphism.

2. Basic

_properties

In this section we will

study

the basic

properties

of the ECF. In _many

ways it resembles the

RCF,

but there are also some _open

questions

which

suggest

that there are fundamental differences.

Let x E

(0, ),

and define

From definition

₍₁₎

_{of TE}

it follows that

where

_{T# (z ) = z}

and =

for n > 1,

and _one has-similar

As usual the

convergents

are obtained via finite

truncation;

The finite continued fraction in the left-hand side of

₍₅₎

is denoted

_by

[[0;

b1,... ,

bn]].

It is clear that the left-hand side of

₍₅₎

is a rational

number,

so that we

have the

_following

result.

Theorem 2.1. Let x E

(0, 1),

then x has a

finite ECF-expansion

(i.e.,

= 0

for

_some n >

1)

if

and

if

_x E

Q.

Proof.

The necessary condition is obvious since x in

(5)

is a rational

number. For the sufficient

_condition,

let x =

P/Q

with

P, Q

E N and

0 P Q. Then

where

bl

=

LQ/PJ.

It is clear that

0 f

Q -

b1P

P and

1 ~

Q

because

_Q

=

b1P

+ _r with

0 ~

_r P

by

the Euclidean

algorithm.

Now let

then

Since

p(N)

E N U

₁₀₁

there exists an n, > 1 such that

_TE(x)

= 0. Notice

that one has to

apply

TE

to x at most P times to

get

TË(x)

= 0. 0

The

_proof

of the

_following

theorem is

_omitted,

since it is

_quite

straight-forward. For a

proof

the interested reader is referred to Section 1 in

[K],

where a similar result has been obtained for a

general

class of continued

fractions.

Proposition

2.1. Let the sequences and

(Qn)n>o

recursively

be

Then and are both

increasing

_sequences.

Furthermore,

one has

for

n > 1 that

-

-and

For the

_{RCF-expansion}

it is known that even-numbered

_convergents

are

strictly increasing

and odd-numbered

_strictly

_decreasing

and that _every even-numbered

_convergent

is less than _every odd-numbered one. The same

result also holds for ECF

_convergents

as stated in the

following

proposition.

Proposition

2.2. Let

_Cn

=

Pn/Qn

be the n-th

ECF-convergent of

_x E

converges to the number from which it is

generated.

Proposition

2.3. For x E

_~0,1),

let be the _sequence

_of

ECF-convergents

of

x. Then

Pn/Qn =

x.

-Proof.

In case x is

rational,

the result is

clear;

see Theorem 2.1. Now

In

_fact,

if x is rational the

_special

case

Tp(x)

= 0

gives

x =

Pn~Qn.

From

from which

For the

_{RCF-expansion}

one has the theorem of

Lagrange,

which states

that x is a

quadratic

irrational

(i.e., x

E and is a root

of ax2 + bx + c

=

0,

a,

b,

c E

Z)

if and

only

if x has a

RCF-expansion

which is

ultimately

(that

is,

from some moment

_on)

_periodic.

For the ECF the situation is

more

complicated. Suppose x

E

~0,1)

has a

periodic

ECF-expansion,

_say

where the bar indicates the

_period.

_Clearly

one has that is a

quadratic

irrational,

and from

₍₄₎

it follows that the

_{period-length t}

is

_{always equal}

to 1.

Due to

_{this, purely periodic expansions}

can

easily

be

characterized;

for n E N one has

One could wonder whether every

quadratic

irrational x has an

eventually

periodic ECF-expansion.

In

_[Ry2],

_Ryde

showed that a

quadratic

irrational x has an

eventually periodic MNK-expansion

if and

only

if a certain set of conditions are satisfied. In Section 4 we will see that the

ECF-map

TE

and a modified version of the

MNK-map

are

isomorphic,

and due to this we

will obtain that not _every

_quadratic

irrational x has an

eventually periodic

ECF-expansion.

An

_{important question}

is the relation between the

_convergents

of the

RCF and those of the ECF. Let x E

[0, 1) B

Q,

with

_{RCF-expansion}

x =

then the

_question

arises whether

_infinitely

_many

_{RCF-convergents}

_and/or

mediants are _among the

ECF-convergents,

and

conversely.

mediants.

A related

_question

is the value of the first

_point

in a ’Hurwitz

spectrum’

for the ECF. Let x E

[0, 1) B Q,

again

with

RCF-convergents

and

ECF-convergents

where we moreover assume that

1 for n > 0.

_Setting

for n > 0

one has the classical results that

which

_{trivially implies}

that

infinitely

often for all x E

(0,1)

B

Q.

If

_infinitely

many

RCF-convergents

of x are also

ECF-convergents

of _x,

then one has that

(14)

C

_{infinitely often,}

with C = 1. Consider the number x

having

_a

purely periodic expansion

with bn

= 2. Then x

= !( -1 + v’3).

The difference

equation

An

= +

2An-2

controls the

_growth

of

_{QnlQnx -}

_Pnl.

Its

_eigenvalues

are

1 -

y’3

and 1 +

y’3,

and therefore we see that

8n(x) =

[

is

asymptotically equal

to 21. Furthermore from

₍₉₎

one can see that

bn+1+2

1 b

Pni

which shows that such a constant C

cannot exist for all x.

3.

_Ergodic

_properties

In this section we will show that

TE

has no finite invariant _measure,

equivalent

to the

_Lebesgue

measure

A,

but that

TE

has

infinitely

_many

a-finite,

infinite invariant measures. Furthermore it is shown that

TE

is

Let

and define for n E

_N,

_{bl, ... ,}

_bn

E N with ...

_bn

the

_cylinder

sets

(or:

fundamental

_{intervals) B (bi, ... , bn )}

_by

Then it is clear

_that,

see also

Figure

2,

Now let _p be a finite

TE-invariant

_measure, that

is,

for _any Borel set A E

(0,1).

Since _it is a _measure, we have that

where _Tn E

B(n)

denotes the invariant

_point

under

_TE,

see also

(13).

On

the other

_hand,

because _IL is

_{TE-invariant.}

_Hence,

since we assumed that _ti is a finite

measure,

Furthermore,

we also have

Similar

_arguments

_yield

that

_{~([3/5, ri]) = p([8 /13, Ti]),}

and therefore

so that we have

Continue this iteration to see that

B(1)

must have its mass

con-centrated at Ti.

Next,

it follows from

₍₁₆₎

that

TE1B(2) _

_(1/3,3/8]

U

(2/3,3/4].

° But

p,(2/3, 3/4]

= 0 so that

~(T~B(2)) =

¡.¿((1/3, 3/8])

and

fol-lowing

the same

arguments

as above

gives

that

B(2)

has its mass

/~(.B(2))

Applied

to other values of n, induction

yields

that on

(0, 1)

the measure

p has mass

p(B(n))

concentrated at Tn for n > 1.

Consequently,

we have

proved

the

_following

result.

Theorem 3.1. There does not exist a non-atomic

finite

TE-invariant

mea-sure.

Next,

we will _prove

ergodicity

of

TE

with

respect

to

Lebesgue

measure.

Theorem 3.2.

_{TE is}

_ergodic

with

_respect

to

_Lebesgue

measure A.

Proof.

Let

TE lA

= A be _an invariant Borel set. Define

where cA denotes the indicator function of A. Then we calculate

We

_put

Note that that

and that it follows from

₍₈₎

and

₍₇₎

I .

From this it follows that

and that

which shows that

For n = 1 _we have _{a more}

precise

estimate. Since

we

_get

Together

with

₍₁₈₎

this

_yields

that

Furthermore we have that

Note that for

_Engel’s

series

_6(b)

=

d(b)

which fact makes the

proof

easier.

Together

with

_(B)

we

_get

the estimate

d(b

+

₁₎

The

_Martingale

_Convergence

Theorem shows that

almost

_everywhere,

see also Theorem 9.3.3 in

[S3].

If = 1 _a.e. there is

nothing

to show.

Let us therefore assume that

a(A)

1.

Suppose

that there is some z E Ac such that = 0. Then for n

sufficiently

large

8(bl (z), ... , b~,(z))

for _any

_given

e >

_0,

and

by

(A)

for b =

bn(z)

_we

find

_d(b)

_{Applying (D)}

this

_yields

that

_{d(c) 4}

for all

_{c >}

b. Since the set

_Fnr

=

N, j >

1}

is countable

(since

_every

x E

FN

ends in a

periodic

ECF-expansion

with

digit

b

N),

it follows that

FN

has measure 0 and we

clearly

have

bn

= _{oo a.e.} Therefore

by

(A)

we see that

6(bi (z) , ... , bn(x)) ~

for almost all

points

x.

Assuming

that e 1 this shows that

_cA(x)

= 0 almost

everywhere,

i.e.,

A(A)

= 0. 0

For the

_Engel’s

series

_R6nyi

_[R]

showed that for almost all x the sequence of

_digits

is

_{monotonically increasing}

from some moment

_no(x)

on. For the

ECF a similar result holds.

Proof.

Setting y

:=

Q.

it follows from

(17)

and

(6)

that

and that 0 _y

1

If we

put

bn

=

bn+l

_we

immediately

get

Lemma 3.1.

Proof

(of

the

_lemma,

see also

[S3],

_p.

68-69).

It follows from

(19)

that

equals

Therefore we have to estimate the sum

For b = _a the first term while for b > _a + 1 _{we use}

0

y C a

to obtain the estimate

and it follows that

we

eventually

get

This

_expression

has its maximum for a = 1 which

gives

the value

313

The claim on the maximal value can be seen as follows.

The sum of the four terms

decreases to 0 as a - oo and becomes smaller

than .1

for a > 3. On the

other hand the

_remaining

term

is

_increasing

and is bounded

_by

_2.

Therefore numerical calculations suffice for n =

1, 2, 3.

This _proves the Lemma.

Now the Borel-Cantelli lemma

_yields

that the set of all

_points

x for which

=

bn+1(x)

for

infinitely

_many values of _n has _measure 0. 0

Now we

give

two constructions of

a-finite,

infinite invariant measures for

TE.

The first construction follows to some extent Thaler’s construction

from

_[T]

of

_Q-finite,

infinite invariant measures for the

Engel’s

series _map

S.

First,

let

_B(n)

be as in

(15).

Define

Obviously

one has that

A(n, k) fl A(m, e)

=

0

for

(n, k) #

(m, e)

and

that,

apart

from a set of

Lebesgue

measure zero

We now choose a

monotonically increasing

_sequence of

positive

real

num-bers

_satisfying

For _any

_{non-purely periodic}

x E B =

B(n),

there exist

positive

in-tegers nand i

such that x E

A().

For _any

non-eventually periodic

and

For x E

[0, 1) B Q

given

one

has,

since

and for k > 2 one has

ax+ux) -

=

n(TE(x)

+

1)2

(ak

(TE (z) ) -

_O:k-l

(TE (z) ) ) .

So

_by

induction it follows that the sequence is a

positive

mono-tonically

non-decreasing

sequence for each x E

[0,1).

We will show that

and

_h(x)

:= 0 for x

fI.

is a

density

of a measure

equivalent

to

Lebesgue

measure, that

is,

it satisfies Kuzmin’s

equation,

see also

Chapter

13 in

_[83].

First note that for x E

B(n)

Kuzmin’s

equation

reduces to

where Vj

is the local inverse

of TE

on

B(j),

i.e.,

As in Thaler’s case for the

Engel

Series

expansion,

this construction

yields

infinitely

many different

cr-fmite,

infinite invariant measures which are not

multiples

of one-another.

For the second

_{construction,}

let Note that

Here Note that _go is a

solution of the functional

_equation

Setting

for t =

2, 3, ...

then

_h(x)

:=

gt- 1 (x)

on

B(t)

is an invariant

density.

To see this note that

Kuzmin’s

_equation

reads

on

B(k).

Then for x E

B(k)

we have on the left

hand,

while

expanding

the

_right

hand side

_gives

In the calculation we used that =

We end this section with a theorem on the renormalization of the

ECF-map

TE.

See also Hubert and Lacroix

[HL]

for a recent survey of the ideas behind the renormalization of

_algorithms.

Setting

. , - I

..-then

_clearly

0 1. We have the

_following

theorem. Theorem 3.4. Let ₇ =

324,

then

Proof.

We introduce for t E

[0, 1]

the _map

Then where

On the other hand we

have,

see also

(17)

Therefore

which

_yields

that

and

Remark.

_Following

the ideas in

_Schweiger

_[S2]

it should be _easy to _prove

that the sequence is

uniformly

distributed for almost all

points

x.

4. On

_Ryde’s

continued fraction with

_{non-decreasing digits}

In

_1951,

_Ryde

_[Ryl]

showed that _{every x} E

(0, 1)

can be written as a

monotones,

nicht-abnehmenden Kettenbruch

_(MNK)

of the form

which is finite if and

_only

if x is rational.

However,

and this was

already

observed

by Ryde,

one has that

and therefore we

might

as well restrict our attention to the interval

(2,1),

and

_just

consider the _map

_{TR :}

(1, 1)

-

(~ 1),

given by

see also

[S3],

_p. 26. Now _{every x} E

(- 1)

has a

unique

NMK of the form

(20)

(with s

=

1),

which _we abbreviate

by

The

_following

theorem establishes the relation between the ECF and the MNK. We omitted the

_proof,

since it follows

_by

direct verification.

Theorem 4.1. Let the

_{bijection 0 :}

_{(0, 1)}

~

(~? 1)

be

defined by

Then

Furthermore,

for

and

_if

we

defines

1 the

cylinders of

TR

by

then

Due to Theorem 4.1 we can

’carry-over’

the whole ’metrical structure’

of the ECF to the MNK. To be more

precise,

letting

x3 be the collection of

Borel sets of

( 2

_1),

and

_setting

where p is a

a-finite,

infinite

TE-invariant

measure on

(0,1)

with

density

h

(with

h from Section

_3),

then we have the

following

corollary.

Corollary

4.1. The _map

_TR

is

_ergodic

with

_respect

to

_Lebesgue

measure

Proof.

We

_{only give}

a

proof

of the first statement.

Suppose

that there

the statement of his theorem covers almost 2

pages!-has

been satisfied.

Due to Theorem 4.1 these constraints can

trivially

be translated into a set

of constraints for the ECF.

As a _consequence there exist

(infinitely many)

quadratic

irrationals x for

which the

_{ECF-expansion}

is not

_{ultimately periodic.}

We end this paper with an

example.

Example.

Let x

= ) ( - 1 + 2U$)

= 0.6944271.... Then the

RCF-expansion

of x is x =

(0;

1, 2, 3, 1, 2, 44, 2, 1,

3, 2, 1, 1, 10,

1 J.

Using

MAPLE we obtained the first 22

partial quotients

of the

ECF-expansion

of x:

Acknowledgements

We would like to thank the referee for several

_helpful

comments and

_suggestions.

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Yusuf HARTONO, Cor KRAAIKAMP

Delft _University of _Technology and Thomas _Stieltjes Institute for Mathematics ITS _(CROSS)

Department of _Mathematics, Universitat _Salzburg Hellbrunnerstr 34

A 5020 _Salzburg

Austria