On inhomogeneous diophantine approximation with some quasi-periodic expressions, II

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

T

AKAO

K

OMATSU

On inhomogeneous diophantine approximation with

some quasi-periodic expressions, II

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{2 (1999),}

p. 331-344

<http://www.numdam.org/item?id=JTNB_1999__11_2_331_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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

On

_{inhomogeneous}

_{Diophantine approximation}

with

some

quasi-periodic

expressions,

II

par TAKAO KOMATSU

RÉSUMÉ. On s’intéresse aux valeurs de

M(03B8,~)

=

lorsque

03B8 est un réel _ayant un

développement

en fraction continue

quasi-périodique.

ABSTRACT. We consider the values

_concerning

M(03B8, ~)

=

~~

where the continued fraction

_expansion

of 03B8 has a

_{quasi-periodic}

form. In

_particular,

we treat the cases so that each

_{quasi-periodic}

form includes no constant.

_Furthermore,

we

_give

some

_general

conditions

_satisfying

_{M (03B8, ~)}

= 0.

1. INTRODUCTION

Let 0 be irrational

_{and 0}

real. We _suppose

_throughout

that is

never

integral

for _any

_integer

_q. Define the value of the function

which is called

_{inhomogeneous approximations}

constant for the

_pair

_{0, 0.}

It is convenient to introduce the functions

Then

_M(8,cP)

= Several authors have treated

M (0, q5)

or

M+(8, Ø)

by

using

their own

algorithms

(See

[1], [2], [4], [5],

[11]

e.g.),

but it has been difficult to find the exact values of

_{M (0,}

_Ø)

for

the concrete

_pair

of 8 and

_~.

For

_{example, Cusick,}

Rockett and Szusz

_([2])

obtain

when 0 =

(1 +

~)/2

=

(1;1,1, ... ~.

And author

([5])

obtains

when 0 =

( a2 + 4 - a)/2

=

[0;

_a,

a, ... ~.

However,

it is not _easy to

apply

these methods to find the value

_{.Nl(e, ~)}

about the different

_types

of 8. In

_[6]

author establishes the

_relationship

between

_{,~l~l (8,}

_Ø)

and the

algo-rithm of

_Nishioka,

Shiokawa and Tamura. If we use this

_result,

we can

evaluate the exact value of

_{M (0,}

_Ø)

for _any

_pair

of 0

_{and 0}

at least when 0 is a

positive

real root of the

_{quadratic equation and 0}

E

Q(8).

For

_example,

are

_given

when 0 =

( ab(ab +

4) -

ab) / (2a) = [0;

_a,

b,

_a,

b, ... .

Furthermore,

in

_[7]

author is so successful

applying

the

Nishioka-Shiokawa-Tamura

_algorithm

that the exact value of

_{M (0, 0)}

can be calculated even

if 0 is a Hurwitzian

number,

namely

its continued fraction

expansion

has a

_{quasi-periodic}

form. And it is the first time to find a concrete

_pair

of 8

and 0

so that = 0. For

example,

for a

_{positive integer s}

is

_given.

In this _paper we consider the cases so that each

quasi-periodic

form

2. NST ALGORITHM

We first introduce the NST

_algorithm

_{(~9~). 8}

= _{a2, - - -}

] denotes

the continued fraction

_expansion

of

_9,

where

The k-th

_convergent

=

ao;

at, ... , ak ~

]

of 6 is then

given by

the

recurrence relations

Denote 0

=

0

[

bo; bl, b2, ... , ~

]

be the

expansion of q5

in _terms of the

sequence

~80, ~1, ... },

where

Then, 0

is

_{represented by}

where

_Dk

=

(-1)~‘6oB1...

8k. Now,

the

following

theorem is

established in

_(6~.

Theorem 1.

where

_{Bn =}

_:E~==1

Remark. It is also known in

_[6]

that +

_~~~

= and ₊ _

(1 -

Together

with

_J1~1+(B,~)

=

M-(C,1 - ~),

_{one can}

obtain the value

_{~l~t(B, ~).}

3. THE CASE

_{,M~B, ~~}

= 0

Continued fraction

_expansions

of the form

are called Hurzuitziarc if _co is an

integer,

_{cl, ... ,} are

_{positive integers,}

...,

Qp(k)

are

polynomials

with rational coefficients which takes

is not constant.

_Ql(k),

... ,

Qp(k)

are said to form a

quasi-period.

The

expansions

where s is a

_{positive integer}

with s &#x3E;

2,

are well-known

_examples

(See

[3],

[8],

[10]

e.g.).

In

_[7]

for

_{a positive integer s}

we have

M( e1/8,

(el/8 -1)/2)

=

0,

M(e’/-’, 1/2)

=

1/8

and

M(eI/8,

1/3)

=0if~=2

(mod 3); 1/18

otherwise.

Then,

what is the condition such that

_{,Jt~l (9, ~) -}

0 holds? It seems

that a non-constant

_polynomial

in a

quasi-periodic

_part

influences whether

M (0, Ø)

= 0 _{or not.}

So,

_we consider the cases each

_{quasi-periodic}

form

includes no constant.

where s is a

_positive

_integer,

or

where s is an odd

_{positive integer}

with s &#x3E;

3,

is one of the well-known

examples

(See [10]

e.g.).

In _any of two

_expansions

of 0 above ak is

increasing

and ak - oo

(k

-oo).

So,

one _may be

_apt

to

_conjecture

that

_{Jl~l(8, ~)}

= 0 for almost all of

0.

But,

there is a case

satisfying

.M(~,~) 7~

0.

Theorem 2.

x B

Proo f .

First,

note that in the

_expansion

of ~ I

yielding

It is convenient to see that

and

Hence,

and

yielding

that

_{M - (0, q5)}

=

1/4.

Next,

1 - ~ =

(1 - 0)/2

=

1/(e1~9

₊

1)

is

expanded

as

and

In a similar _manner,

_by

’1

Contrary

to this

_result,

there

_is,

of _course, a case

satisfying

M(0,

§)

= 0.

Theorem 3.

Remark. It is

_interesting

to see that in

[7]

, - , B.

in

_comparison

with Theorem 2 above and this Theorem.

Proof. 0

=

1/2

is

expanded

_as

one finds that

We shall show one more case

satisfying

,NI (8,

§)

= 0.

Theorem 4.

/ -, "

Proof.

When s - 0

_{(mod 3), ~}

=

1/3

is

expanded

as

as n tends to

_infinity.

_Hence,

we have

M - (0, 1/ 3)

= 0.

1 -

_ql

=

2/3

is

expanded

as

as n tends to

_infinity.

_Hence,

we have

M+(9,

1/3)

= 0.

When 5=1

_{(mod 3), ~}

=

1/3

is

_expanded

_as

as n tends to

_infinity.

Since for n

_{=1, 2, ...}

one finds that

i a.__1I

as n tends to

_infinity.

_Hence,

we have

M - (0, 1/3)

= 0.

as n tends to

_infinity.

Since for n =

_{1, 2, ...}

1

one finds that

as n tends to

_infinity.

_Hence,

we have

.M+(~ 1/3)

= 0.

Therefore,

M(0, 1/3)

= 0. When s - 2

_{(mod 3), ~}

=

1/3

is

_expanded

as

as r~ tends to

_infinity.

Since for n =

1, 2, ...

one finds that

and

_4&#x3E;0

=

1/3,

for n =

1, 2, ...

as n tends to

_infinity.

Since for n =

1, 2,

...

one finds that

as n tends to

_infinity.

_Hence,

we have

M+(0,

1/3)

= 0.

Therefore,

M (0, 1/ 3)

= 0.

Let us calculate

_{M (0, 0)}

when

where s is an odd

_positive

integer

with s &#x3E; 3. The situations are a little

bit different from the

_previous

results. Notice that an =

(2n -

1)s

~ _oo,

so

On-1 =

1/(an

+

_{8n) -7 0 (rt}

=

1, 2, ...

_-

oo).

I

= 1

and

_{qn-, IDn-1 I}

= 0 hold for this 8 _too. The first result is

_quite

different from Theorem 2.

and

Since for n =

_{1, 2, ...}

one finds that

and

Since for n =

1, 2, ...

one finds that

Proof. 0

=

1/2

is

expanded

as

and

as n tends to

_{infinity. Therefore,}

we have

M(~, 1/2) = M±(O, 1/2)

= 0. 0

Theorem 7. , ,

-

,-with

_{s 0, s 1, s - 2}

_{(mod 3),}

_{respectively.}

0

5. SOME CONDITIONS SATISFYING

cp)

= 0

We have

_already

seen several

examples

so that

M (0, q5)

= 0 holds.

Then,

what is the condition of

_M(0,

_cp)

= 0? Of _course, the

following

is clear.

Theorem 8.

_If

_{§n -}

0 or 1

(n

-3

_{f or infinitely}

_{man y}

_positive

integers

n, then = 0.

Proof.

First,

we shall show that

BnlDn-11

[

4 for _any

positive

we obtain

On the other

_hand,

’--’

Corollary.

When bn =1,

On-, -+

0 i f

and 1

_{(n -}

_oo).

This is _{very generous.}

_So,

we state the

_following.

Theorem 9.

_If

_{I an -}

c and an -+ 00

(n

-~

_oo)

_{f or infinitely}

_many

positive

integers

n, then

M (0, §)

= 0.

_Here,

_c is _{a constant not}

_depending

npon n.

Remark. In

_fact,

an =

bn

-~ oo

(n

-

_oo)

holds in all

_previous

theorems above

_implying

_{JI~! (8,}

_Ø)

= 0.

0

REFERENCES

[1] J. W. S. Cassels, Über limx~+~x|~x + 03B1 2014 _y|. Math. Ann. 127 (1954), 288-304.

[2] T. W. _Cusick, A. M. Rockett and P. Szüsz, On _{inhomogeneous Diophantine approximation.}

J. Number _Theory 48 _(1994), 259-283.

[3] C. S. Davis On some _simple continued fractions connected with e. J. London Math. Soc.

20 _(1945), 194-198.

[4] R. Descombes Sur la répartition des sommets d’une ligne polygonale régulière non _fermée. Ann. Sci. École Norm Sup. 73 _(1956), 283-355.

[5] T. Komatsu On inhomogeneous continued fraction expansion and _{inhomogeneous}

Dio-phantine approximation. J. Number Theory 62 _(1997), 192-212.

[6] T. Komatsu On _{inhomogeneous Diophantine approximation} and the Nishioka-Shiokawa-Tamura algorithm. Acta Arith. 86 _(1998), 305-324.

[7] T. Komatsu On _{inhomogeneous Diophantine approximation} with some _{quasi-periodic}

ex-pressions. Acta Math. _Hung. 85 _(1999), 303-322.

[8] K. R. Matthews and R. F. C. Walters Some _{properties of} the continued _{fraction expansion}

of

_(m/n)e1/q.

Proc. _Cambridge Philos. Soc. 67 _(1970), 67-74.

[9] K. _Nishioka, I. Shiokawa and J. Tamura Arithmetical _{properties of} a certain power series. J. Number _Theory 42 _(1992), 61-87.

[10] O. Perron Die Lehre von den Kettenbrüchen. Chelsea _reprint of 1929 edition.

[11] V. T. Sós On the _theory of Diophantine approximations, II. Acta Math. Acad. Sci. _Hung.

9 _(1958), 229-241. Takao KOMATSU Faculty of Education Mie _University Mie, 514-8507 Japan E-mail : _{komatsuQedu.mie-u.ac. jp}