The metric simultaneous diophantine approximations over formal power series

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

K

AE

I

NOUE

The metric simultaneous diophantine approximations

over formal power series

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 151-161

<http://www.numdam.org/item?id=JTNB_2003__15_1_151_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

The metric simultaneous

_Diophantine

approximations

over

formal

_power

series

par KAE INOUE

RÉSUMÉ. Nous étudions les

_{propriétés métriques}

de

l’approxima-tion

_{diophantienne}

simultanée dans le cas non archimédien. Nous

prouvons d’abord une loi du 0 - 1 de _type

Gallagher,

_que nous

utilisons _{ensuite pour} obtenir un résultat de _type Duffin- Schaeffer.

ABSTRACT. We discuss the metric

_theory

of simultaneous

Dio-phantine approximations

in the non-archimedean case.

First,

we

show a

_Gallagher

_type 0-1 law. Then

_by

_using

this

_theorem,

we

prove a Duffin-Schaeffer _type theorem.

Introduction

In the

_study

of metric

_{Diophantine approximations,}

one of the

questions

asks for conditions on a

_{non-negative function 1/J}

which

_guarantee

that the

inequality

I I a" ’"

has

_infinitely

many

solutions R

for almost every x. In the one-dimensional q

case it is known that if the

_following

two conditions hold then

₍₁₎

has

infinitely

many solutions for almost every x, Duffin-Schaeffer

[1]:

where c is a

_positive

constant

_{and 0}

is the Euler function.

A similar result holds for s-dimensional

_{Diophantine approximations.}

In the

_previous

_paper

_[4],

we discussed an

_analogue

of Duffin-Schaeffer theorem for formal Laurent _power series

_{(one-dimensional case).}

As its continuation we consider the same

_question

for

_higher

dimensional case in this _paper. To state our results we start with some fundamental definitions and notations.

We use the

following

notations:

F : a finite field with _q

_elements,

the set of

_polynomials

with

_{F-coefficients,}

F(X) :

the fraction field of

IF((X-1)) :

the set of formal Laurent power series.

We define the absolute value of

_f

E

_{lF((X-1 ))}

_by

_If

-

qdegf

and

_put

which is a

_compact

abelian group with the metric

d( f, g )

=

f - 91.

We

denote by

m the normalized Haar measure on L. This

implies

that for any

In

_[4],

we showed that

has either

_infinitely

_many solutions for m-a.e.

_f

E L or

_{only finitely}

_many solutions for m-a.e.

f

E L

_{Here, 0}

is a

In

&#x3E;

0}

U

101-

valued function and P and

_Q

have no non-trivial common factor. We call this

"Gallagher

type

theorem" since this

property

was

proved by

Gallagher

in the classical case. Then, we showed "Duffin-Schaeffer

_type

theorem" :

Let o

be a

0}

U

_(0)-valued

_function,

which satisfies

Suppose

for a

positive

constant

_C,

there are

_infinitely

_many

_{positive integers}

n such that

holds,

then

The _purpose of this paper is to show the s-dimensional version of these theorems for formal Laurent power series

(s &#x3E; 2).

For a

given V)

as

above,

we consider the

following inequalities:

We shall

_give

a sufficient condition

on 9

so that

(3)

has

infinitely

_many

solutions

_{( pl , ... , p )}

for almost _every

_{( f 1, ... ,}

_fs)

E ILS with

_respect

to

m~,

which is the s-fold

_product

of m. For this _{purpose, let}

for a monic

_{polynomial Q}

and

we first _prove the

following

theorem in

§1.

Theorem 1. For

_{either ms(E)}

= 0 _or 1

holds, equivalently,

(3)

has

infinitely

many solutions

of

(Q,

Pl, ... ,

P,)

for

m~-a.e.

( fl, ... , f,)

E L~ or has

_{only finitely}

_many solutions

_for

_{(fl,... ,}

_f,)

E L~.

We call this the s-dimensional version of the

_Gallagher

type

theorem.

Secondly, making

use of this

_theorem,

we can show the s-dimensional

ver-sion of the D uffin- Schaeffer

_type

theorem in

_§2.

Theorem 2. be a

01

U which

_satisfies

Suppose for

a

positive

constant

C,

there are

infinitely

_many

positive integers

n such that

holds. Then

₍₃₎

has

_infinitely

solutions

~ , ... , p-

_,for

( f p ... , f s ) E IL.~s .

Finally,

we

_explain

a Duffin-Schaeffer

_type

_conjecture

in formal Laurent

1. Proof of Theorem 1

In the

_sequel,

we

_study

the metric

_property

of s-dimensional simultaneous

approximations

(3)

for s &#x3E; 2. For

_f

= _E

F((X-1))

(al

#

0),

we

_put

For

_{given hi}

E such that

we define the

_cylinder

set ... ,

his)

as follows:

Then we see the

_following,

...

Lemma 1. Let

_{I (hl k}

_{h2 k ,}

... ,

hs k) k

&#x3E;

1}

be a sequence

of cylinder

sets

defined

as above with

11

be a _sequence

_of

measurable sets

_{for which Ek}

C

_{( hi k ,}

... ,

hs k ) .

Suppose

there exists 6 &#x3E; 0 such that &#x3E;

hsk))

for

any k

&#x3E;

1. Then

Proof.

Let

We show that ml

_(Ht)

= 0 for

any l &#x3E;

1,

which

_implies

the assertion of this lemma.

_Suppose

there exists a

I~o

E

_Z+

such that &#x3E; 0

_ko.

There is a natural

_{correspondence}

between

_cylinder

sets defined for L as in

(2)

and

_q-adic

rational

_intervals,

and so

_{(L, m)}

is

_isomorphic

to

_{[0, 1]}

with the

_Lebesgue

measure.

_Similarly,

mS)

is

_isomorphic

to with the

Lebesgue

measure. So

by using cylinder

sets

_{(hl, ... , hs)}

C L~ instead of

h

x ... x

_Is

C

_[0,

we can

_apply

_{Lebesgue density}

theorem. Then we

for some l~. On the other

hand,

From the

_assumption

of this

lemma,

That is

which contradicts

_(4).

0

Lemma 2. For _any

_{polynomial hi}

E

₀₎

and _gi E 1 i _s, the _map T

_of

ILS onto

_itself

_defined

_by

Proof.

It is easy to see that each _map

is a Bernoulli transformation of L. In other

words,

if we

_put

1 ~

gives

a _sequence of

independent

and

identically

dis-tributed random variables. In

_particular,

Ti

is weak

_mixing.

Since the s-fold

_product

of weak

_mixing

transformations is

_ergodic

_(see

_[5]

_Prop.

4.2.),

this

_yields

the assertion of the lemma. 0

Proof of

Theorem 1. In the case of formal _power

series,

we mean

_by

(Pi, Q)

= 1

that Pi and Q

have _no non-trivial _common factor.

If

then we can find a _sequence of

polynomials Qi , Q2, Q3,

... and a

positive

integer

l, such that

_O(Qk)

&#x3E;

q-l

for _any k &#x3E; 1. In this _case, for any

fi E

IL and a

_{sufficiently large I~,}

we can find

_Pi

_(deg

_Pi

_deg

Qk),

such that

and Pi

and

_Qk

have no non-trivial common factor.

Indeed,

the number of

polynomials

P

such that

is all such

_{polynomials P}

are not

_{relatively prime}

to

_~?~,

then

_Qk

has more than

qdeg Q k -l

factors,

which is

impossible

if

deg Q k

is

sufficiently large.

This

_implies

E

Now we show the assertion of the theorem when

For fixed

_{Q,1’1, ... ,}

_PS-1

and

Ps,

there exist

_polynomials

_{hl, ... , hs}

such

that Then

₍₆₎

_implies

deg h2

deg

tends

to oo. Thus we can

_apply

Lemma

1 when

₍₆₎

holds. Then we evaluate the measure of

_{Udeg Q&#x3E;n}

E’Q’

Let

R be an irreducible

_polynomial

and consider

for

We

_put

and

Then we see

and

for j

=

0,

1. From Lemma

1,

we find that

for

_{( fl, ... , fs)}

E L .

_Suppose

_(7),

we have

and see

Also,

we have

Here,

These

_imply

for j

=

0,1.

Hence from Lemma

2,

_we have

for j

=

0,1.

Thus,

if either

_R))

or

ms(Ei(1 :

R))

&#x3E; 0 for some

irreducible

_{polynomial R,}

then we have

m (E)

= 1.

Now we assume that

R))

=

R))

= 0 for

any irreducible

_polynomial

R. We

_put

where

₍₈₎

refers to:

Suppose (8),

we have

for any

polynomial

U with 0

deg

U

deg

R.

Here,

we see

which

_implies

that if

then its measure is

qs deg R

Since

F(R)

is

_(.

+

~)-invariant,

_{(0 deg U}

_degR)

and

{(11, ... ,

1I.f

_I

_{deg f}

i

- deg R~

have the same measure. Hence we have

which

_implies

Suppose

ms(E)

&#x3E;

_0,

since E =

F(R)

U

_{Eo(I, R)}

U

_El(1,R),

we see that

mS(F(R))

&#x3E; 0 for _any irreducible R.

_By

the

_{density theorem,}

we have

ms (E)

=

m8 (F (R))

= 1 where R is so chosen that

_{deg R}

is

_sufficiently

large. Otherwise,

ms(E)

= 0. 0

2. Proof of Theorem 2

In the

_sequel,

we

_always

assume that

Q , Ql ,

Q’

and

Q’

are monic. Recall

From

_(6),

if is

_{sufficiently small,}

then we have

Now consider the measure of

_EQ

fl

_{EQ, (deg Q’ deg Q).We}

let

_{N(Q, Q’)}

be the number of

_pairs

of

_polynomials

P and P’ which satisfies

for

_{given Q}

and

_Q’e

Then we show that the measure is bounded as

follows,

holds for some

polynomial

R and D =

(Q,

Q’).

If D divides

R,

we _may write

and have

If P* and P’* also

_satisfy

_(12),

then

From

_(12),(13),

we

_get

From

_(14),

we see

which

_implies

that

must hold. Thus the

_possible

number of

_polynomials

P

_satisfying

₍₁₁₎

for

a

_given

R is no more than

q deg D

. Since

(10)

implies

and R is divisible

_{by D,}

we find that

Then

Because we assume

for

_{sufficiently large deg Q.}

From

₍₄₎

and

_(9),

we have

for

_infinitely

_many n. Then &#x3E;

(2 sC2)-l’

_by

₍₁₅₎

and Lemma 5 of

[8] (pp.

17-18).

Finally using

to Theorem

_1,

we

complete

the

proof

of this

theorem. 0

Example.

Put

Then,

from Theorem 2.2 of

_[7],

we have

and it is easy to see that

holds. Thus we see that there are

_infinitely

_many solutions

(-~-,... ? ~f)

with irreducible of

for a.e.

_{( f 1, ... ,}

e

Remark. It is natural to ask whether we can

_get

a _necessary and suf-ficient condition instead of

₍₄₎

in Theorem 2. In this _sense, we

_give

the s-dimensional Duffin-Schaeffer

_type

_conjecture

in the

_following.

Conjecture.

(3)

has

_infinztely

_many solutions

_{( ~ , ...}

_for

mS -a. e.

(11,... , fs) E

U

_if

and

_{anly if}

diverges.

In the classical _case, the s-dimensional Duffin-Schaeffer

_conjecture

was

proved Pollington

and

_Vaughan

_[6]

for s &#x3E; 2. We _may also _prove this

conjecture

for the s-dimensional formal _power

_series,

s &#x3E;

_2,

if we estimate the lower bound of We will discuss it in another occasion.

References

[1] R. J. DUFFIN, A.C. SCHAEFFER, Khintchine’s _problem in metric _{diophantine approximation.} Duke Math. J. 8 _(1941), 243-255.

[2] G. W. EFFINGER, D. R. HAYES, Additive Number _{Theory of Polynomials} Over a Finite Field. Oxford University Press, New York, 1991.

[3] P. X. GALLAGHER, Approximation by reduced fractions. J. Math. Soc. _Japan 13 _(1961), 342-345.

[4] K. INOUE, H. NAKADA, On metric Diophantine approximation in _{positive characteristic,}

preprint.

[5] M. G. NADKARNI, Basic _{Ergodic Theory.} Birkäuser Verlag, Basel-Boston-Berlin, 1991.

[6] A. D. POLLINGTON, R. C. VAUGHAN, The k-dimensional Duffin and _{Shaeffer conjecture.} Sém. Théor. Nombres Bordeaux 1 _(1989), 81-87.

[7] M. ROSEN, Number _{Theory in} Function Fields. _{Springer-Verlag,} New _{York-Berlin-Heidelberg,} 2001.

[8] V. G. SPRIND017BUK, Metric _{Theory of Diophantine Approximations.} John Wiley &#x26; Sons, New

York-Toronto-London-Sydney, 1979. Kae INOUE Department of Mathematics Keio _University 3-14-1 _Hiyoshi, Kohoku-ku Yokohama, 223-0061 Japan kae@math.keio.ac.jp