Continued fractions, multidimensional diophantine approximations and applications

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

N

IKOLAI

G. M

OSHCHEVITIN

Continued fractions, multidimensional diophantine

approximations and applications

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{2 (1999),}

p. 425-438

<http://www.numdam.org/item?id=JTNB_1999__11_2_425_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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

Continued

_Fractions,

Multidimensional

Diophantine Approximations

and

_Applications

par NIKOLAI G. MOSHCHEVITIN

RÉSUMÉ. Cet article rassemble des résultats _généraux

d’appro-ximation

_{diophantienne,}

sur les meilleures

_{approximations}

et leurs

applications

à la théorie de _répartition uniforme.

ABSTRACT. _{This paper is} a brief review of some

_general

Dio-phantine results,

best

_{approximations}

and their

_applications

to

the

_theory

of uniform distribution.

1. DIOPHANTINE APPROXIMATIONS.

1.1. One-dimensional

_{approximations.}

1.1.1.

_Lagrange

_spectrum.

Let a be an irrational number. Dirichlet’s theo-rem states that there are

infinitely

_many

positive

integers q

such that

holds,

where I I - I

I

denotes the distance to the nearest

_integer.

Hurwitz obtained a more

precise

result: for _any irrational number _a, the

inequality

has

_infinitely

_many solutions in _q.

_Moreover,

there is a countable set of numbers a for which this

_inequality

is an _{exact one,} that

_is,

for _any

positive

c there are

_{only finitely}

_{positive integers q}

such that the

inequality

holds.

We define the

_Lagrange

spectrum

to be the set of the real numbers A for which there exists a =

a(A)

such that the

inequality

has

_infinitely

_many

_solutions,

and for any

positive c

the

inequality

’1

has

_only

a finite number of solutions. It is well-known that

Lagrange

spec-trum has a discrete

_part

i i

y-

y-and the minimal A for which there are

_uncountably

_many a =

a(a)

is

A =

1/3.

Also it is well-known that

Lagrange

_spectrum

contains an interval

[0,A1.

Moreover,

for _any

_{decreasing function 0 satisfying}

_0(y)

=

o(y-1),

_{as y}

tends to

_infinity,

there is an uncountable set of real numbers a such that

the

_inequality

has

_infinitely

_many

_solutions,

but for _{any c} &#x3E;

_0,

the

_stronger

_inequality

has

_only

a finite number of solutions.

One can find the above results in

[5].

All of them can be obtained from

the continued fraction

_expansion

_[14].

1.1.2. Best

_{approximations.}

and continued

_fractions,.

_Any

real number a

may be written as

...

...

where

_{bo E}

Z

_and,

_{for j}

&#x3E;

_0,

_bj

are a

nonnegative

integers.

For

convenience,

we use the notation

a =

This

_{representation}

is infinite and

_unique

when a is irrational. If a is

rational,

we have a =

(bo;

bl, b2, b3, ... ,

bt],

and this

representation

is

unique

if we

_impose

the

condition bt

~

_0,1.

Convergents

to a of the order v are defined as

A

_simple

theorem states that these fraction and

_only

these form the best

approximations,

that is the relation

holds for the

_{denominators q,}

and

_only

for them

_{(see [14J).}

We now

_give

two other _easy facts.

Theorem 1. We have

Ilqllall

~

_(in

order

_of

_{approximation).}

Proposition

2. We have

1.1.3. Klein

_polygons.

We now consider the

_integer

lattice

Z2

C

Jae2.

Let

(q, a)

e Z

be a

_{primitive point}

(gcd(q, a)

=

1)

and _q, a &#x3E; 0. We define the

two

_angles

_cp+ and _p-

_by

I I

B~ ..1

Klein

_polygons

_{K+ (a, q)}

and

_{K- (a, q)}

are defined to be

_respectively

the

following

borders

- - ft

and

which consist of finite

_(nontrivial)

intervals. We now define

0(a,

q)

to be the domain:

We have

Theorem 3

_{([7, 9]).}

1. The vertices

_of

_{7C_ (a, q), (different}

_from

_{(q, a))}

are

integer points

of

the

_form

_(q2p,P2p),

where

_(P2p./q2p)

is the

_2p-th

_convergent

to

_a/q.

2. The vertices

_of

_K:+(a,q)

_(different

_from

_{(q, a))}

are

_{integer points}

_of

the

form

(q2v+l,P2v+l),

i where

P2v+l/Q2v+l

is the

convergent

to

a/q.

3.

_If

_{(u, v)}

E

_{(1C+(a, q)}

U

_{1C_ (d, q))}

fl

7G2

is an

_{integers point}

then

v/u

is a

convergent

to

_a/q

or one

of

the intermediate

fractions

(wp,

+

qv-l),

1 w

_b,+,.

4.

_0(a,

_q)

fl

7Gz

=

0.

One can

easily verify

the same results for infinite continued fractions

(i.e.

for irrational

_numbers).

Recently,

several _papers

_[1,

_{21, 45, 46,}

₂₇₁

devoted to multidimensional

generalization

of Klein

_polygons

have

_{appeared. Unfortunately}

one must

1.1.4.

_{Representation of}

rationals. The rationals

_a/q

with bounded

_partial

quotients

bi

are of

_great

interest

(see [22,

23, 24,

11]).

Let

_{N(k, q)}

be the number of

_{integers A,}

1 A _q,

_{gcd(A, q)}

= 1 such

that _any

_{component bi}

of the continued fraction

_expansion

is bounded

_by

_{k: bi k, i}

=

_{1, ... ,}

n(A).

It is known

([22,

_4,

52])

that

if k &#x3E;

_{q log q}

with ₇

_{sufficiently large,}

then

_{N(k, q) &#x3E;}

1. Moreover we can

show that for almost all

_{positive integers}

q and A with 1 A q, all

partial

quotients

are bounded

_by

O(log q).

By

the _way we _may recall a famous and still _open

_conjecture

which asserts

that for _any

_{q &#x3E; 1,}

we have

N(6, q) &#x3E;

1. However it is known that the

conjecture

holds

_{when q}

= 2Q or _q = 3’

(~39~).

Sergei Konyagin

(see

[17]),

by

means of

Farey fractions, proved

the

fol-lowing

upper bound for

N(k, q):

Theorem 4. For _{any y} 1 and

_for

_any k &#x3E;

_{k( 1)}

we have

mhere cp

denotes the Euler

_function.

We define the sequence

A1

A2

...

_Ad

to be an almost arithmetic

progression

if

’" -

’"

_

In

_[32],

the author shows that numbers with bounded

_partial

_quotients

cannot appear very

regularly: they

cannot form

_long

almost arithmetic

progressions.

The

_following

theorem

_improves

the result from

_{~32~ .}

Theorem 5. For d &#x3E;

_3,

let

_{Ao, ... , Ad}

be

_{positive integers.}

_Suppose

(i)

0

_Ao

...

Ad f orm

an almost arithmetic

progression;

(ii) gcd(Ai, q)

=

1,

i =

0, ... ,

d.

Let

_Av/q

=

[~,1)" - ~

Then there exist vo and ito such that

.i-neorem 5 is _{provea oy} means oi mem _{polygons, me} same result is true for real-valued

_{(not integer)}

almost arithmetic

_progressions

and in the last case S.

_Konyagin

showed that the result for real-valued

_progressions

is

1.2. Simultaneous

_{approximations. Let 0 : 1~ --~ R+}

be a

_positive

and real-valued function. For

_given

a -

(al, ... , as)

E

_W,

a

positive

integer

p is said to be a

0-approximation

of _a, if

1.2.1. Dirichlet and Liouville’s theorems. Dirichlet’s theorem states that for

any a -

(al , ... ,

E

_{R ,}

where

_{1, al , ... ,}

a$

linearly independent

over

Z,

there are

infinitely

_many

0-approximations

of a with

0(y)

=

y-1/3.

On the other

_hand,

Liouville’s theorem

_{([2], ch.5)}

shows that for _any

a =

(al,

... ,

as)

E R’ such that

1,

at, ... , as form a basis of a real

algebraic

field of

_{degree s}

+

1,

there exists

_C(a)

such that

One can see that there are

_{only countably}

_many

_algebraic

a =

(01, ... ,

a~).

1.2.2. Theorem

_by

Cassels and

_Davenport

and the result

_by

Jarnik. In

_{[3, 61}

the

_following

result is obtained.

Theorem 6. There exists a constant

_C,

_for

which there exists an

uncount-able set

_of

elements a which do not have _any

0-approximation

where

v

V. Jarnik

_{[12, 13]}

_proved

another result:

Theorem 7. and A be

_positive

real-valued

_functions

such that decreases as y -&#x3E; oo and

A(y)

-&#x3E; 0 as y - oo. Then there exists an

un-countable set

_of

elements a E R-’

_for

which there are

infinitely

_many

c-approximations.,

but

_{only finitely}

many

OA -approximations.

A review of other results can be found in

[43,

10,

2].

1.2.3. Exact results in termes

_of

the order

_of

_{approximation.}

_Generalizing

the work

_[3]

_by

means of chains of

parallelepipeds

[28,

50, 7,

8]

we

_improve

Jarnik’s result.

Theorem 8.

_{For y &#x3E;}

_1,

and w such that Let

_0(y)

= where

w(y)

decreases as _{y -} oo and

Then there exists a vector a =

(al, ... , as)

which has

infinitely

_many

Theorem 9. Let w

_{and 0}

be as in Theorems 8 and _suppose that

- , -, - - , - . ,

.

Then there exists an uncountable set

of

vectors a =

(a,, - .. , as),

each

of

them

_{having infinitely}

_many

_{0 -approximations}

but not _any

2-d+3-approximation.

It follows that in Cassels Theorem 6 we _may

_put

We _say that a =

(at,... ,

satisfies the

0-condition

if a has

infinitely

many

0-approximations

but not any

c~-approximation

for some c =

c(a)

Theorem 10.

_{Let 0}

be

_{defined by}

_{1/J(y) =}

y-¡/sú)(y)

where cv is

decreas-ing positive

function.

Then in any Jordan s-dimensional domain 0 with Vol 0 &#x3E;

_0,

there exists an uncountable set

of

a E

_satisfying

the

1/J-condition.

Theorems

_{8, 9,}

10 are discussed in

[33].

1.2.4. Successive best

_{approximations.}

Let a =

(al, ... , as)

_E We

de-fine a best simultaneous

approximations

(b.a)

of a to be _any

_integer

_point

, =

(p, aI, ... ,

E such that

_Vq,

_{‘d (bl , ..., b~ )}

1 ~

_q

_p,

(q, b,7 * * * 7 bus) 54 (p,al,... ti°3 ) ,

we have

Let

_{aj 0}

_{Q, j =}

_{1, ... ,}

s . Then all b.a. of a form infinite _sequences

B ~ 1

-

~ ,

For a =

(al, ... , as)

satisfying

aj

ft Q, j

=

1, ... ,

_s, we define

_R(a)

E

[2, s

+

_1]

to be the

_integer

R(a)

= min

In :

there exist a lattice A C

zS+1

with dim A = n

and a natural vo such that

C’

E

_A,

tlv &#x3E;

Proposition

11. Let s = 1. Then

for

_any v &#x3E; 1 _we have det ::l:1

(rank M, [a]

=

2,

t/v).

Proposition

12. For

_{any s &#x3E;}

1 we have

R(a)

=

dimz

(al, ... ,

Proposition

13. Let s = 2 and

al, a2 such that

1, al, a2

are

linearly

in-dependent

over Z. Then

f or infinitel y

_{man y} v we have

Proposition

11 - 13 can be

_easily

verified. The

_following

result is

_proved

in

_[36].

Theorem 14. Let s &#x3E; 3. There exists an uncountable set

of

elements

cx =

(a,,...

such that

(i)

1,

al , ... , are

linearly ind epend ent

over

Z,

(thus dimz (a,, ... , a~,1)

= s +

_1),

and

(ii)

3,

V v &#x3E;

_1,

(Hence

for

all v &#x3E; 1 we have det =

0) .

Theorem 14

_represents

a

counterexample

to the

_conjecture

of J.S.

La-garias

[26].

It shows that the successive b.a. have no such an

_asymptotic

property

as a reader can see in

Proposition

12. The idea of the

proof

was

suggested

to the author

_by

N.P. Dolbilin.

1.3. Linear forms.

_Again,

let _{a1, ... ,} as be real numbers such that

1, aI,

... , are

linearly independent

over

Z,

and

put

a =

For m =

(mo, ml , ... ,

_E

7~ ~+1 ~

101

we define

A vector m E

₁₀₁

is a best

_{approximation}

of

a in sense

of

linear

form

if

All best

_{approximations}

form _sequences

where m, =

(mow, ... , ms,lI)

is the vector of the v-th

b.a.,

(II

(m,)

and

maxj

By

Minkowski’s Theorem we have

_{~’v Mv+ 1}

1.

1.3.1.

_S’ingular

_systems.

The theorem on the order of

_{approximations}

from

§ l.1.2

does not admit multidimensional

_{generalization}

in the sense of linear

form.

Theorem 15

_{(see [29, 35]).}

Let s be an

_{integer &#x3E;}

1

and 0

a

function

such

that

_0(y)

decreases to zero when _y tends to

_infinity.

Then there exists an

uncountable set

_of

elements a =

(al, ... , as)

E 11~~ such that

and

(ii)

the _sequence

_of

the best

_{approximations of}

a

_satisfies

In the case s = 1 this theorem means that there are real numbers with

any

given

order of the best

approximations.

In

higher

dimensions it

gives

something

more.

Khinchin

_[15]

defined a vector a =

(al, - - .,

as )

_E Jae" _to be _a

0-singular

system

if for _any T &#x3E; 0 the

_system

has a nontrivial solution

(mi, m2, ... ,

m$)

E Zs.

Proposition

16.

_System

is

_1/J-singular

4==:t,

_(v

Vv.

1.3.2. Successive best

_{approximations}

_for

linear

_{f orm.}

Here we define

_Ag

to

be the determinant of the successive best

_{approximations}

The

_proposition

below follows from Minkowski theorem on convex

_body.

It seems to me that it is a well-known

fact,

but I could not find the

corre-sponding

reference.

Proposition

17. Let s = 2. Then

for infinitely

_many v we have

A’ 54

0.

The theorem below was

proved by

the author in

[35]

by

means of

_singular

systems.

Theorem 18. Lets &#x3E; 3. Then there exists a uncountable set

of

vectors a =

(al, ... , as)

_E R~ such that

(i)

l, al, ... ,

as are

linearly independent.

over

Z,

and

(ii)

there exists a linear

subspace

£0.

C

_{dim £0. =}

3

_satisfying

the condition

We see that for s &#x3E; 3 almost all best

_{approximations}

_may

asymptoti-cally

lie in a three-dimensional

plane

but

they

cannot lie in two-dimensional

plane.

Of course these

_examples

are

_degenerated

in sense of measure. For

almost all vectors a E R’

_(in

the sense of

Lebesgue)

best

_{approximations}

are

asymptotically

(s

+

_{1)-dimensional.}

2. INTEGRALS FROM _{QUASIPERIODIC} FUNCTIONS

In the text below we discuss

applications

of

diophantine

results to certain

problems

of uniform distribution of irrational rotations on torus. A full review of the methods and results of the

_theory

of uniform distribution is

given

in

_[25].

2.1. Uniform distribution. Let T be the one-dimensional torus and

_{f :}

T’ -7 R defined

_by

the series

We also define the

_integral

where _o;i,... c~9 E R are

linearly independent

over

_Z,

_{and p}

=

(cpl, ... , w.,)

E

H.

_Weyl

_[51]

_proved

that if

_f

is a continuous

function,

then for _{any p} we

have

_I(T,

_cp)

=

o(T),

T - _oo. This

equality

holds

uniformly

in _cp if we

suppose moreover that

f

is smooth.

V.V. Kozlov

_conjectured

that the

_integral

_{I(T, cp)}

is recurrent that is the

following

condition holds:

This

_conjecture

is true when

_f

is any

trigonometric polynomial,

and in this case

_(*)

holds

_uniformly

_{in cp.} This

_implies

that for _any

_{trigonometric}

polynomial f

of finite

_degree,

J°° defined

_by

is itself

_recurrent,

that is

2.2. Case s = 2. In the two-dimensional

case, the

conjecture

above was

proved

by

V.V. Kozlov himself for functions

_f

E

C2(Tz)

in

_[18]

_(see

also

~19~).

It is also _{easy to} see that when

f

is a smooth

function,

then

(*)

holds

uniformly

in cp, that is

_($)

is true. E.A. Sidorov

_[44]

obtained a similar

2.3. The

_general

result. The author

_[34]

_proved

the

_conjecture

in the

general

case:

Theorem 19.

_Suppose

that

belongs

to the class

_Cd(T’),

where d &#x3E;

CS,3

and _{WI, ...,w,} are

linearly

ind epend ent

over Z. Then

f or

_{any p,} the

integral

I(T, Sp)

satisfies

(*).

The

_proof

is based on consideration of best

approximations

in the sense

of linear form

_{(see § 1. 3.2. ) .}

2.4. Metric results. It is known

_{[49, 48]}

that for almost all

_(in

the sense

of

_Lebesgue)

vectors w =

(wl, ... , ws )

_E

R~,

if

f

is smooth

enough,

then

the

_integral

_{I(T, Sp)}

is bounded when T e oo

uniformly

in cp. Hence the

integral

1 (T, c,p)

satisfies

_(*),

_uniformly

_{in cp.} But even in the case s &#x3E;

_3,

this result is not universal.

Let $ be a

decreasing

function and assume that the series

converges. We define a

_periodic

function O : T-’ --7 R to be

_of

the

_{type E}

_if,

the coefficients

_0~i,...,m

in the

_expansion

,V J. ~ ,

The result below is

_proved

in

_[29].

Theorem 20. Let s &#x3E; 3. Then

_for

_any

_{function 4D}

which decreases to zero as _{y -3} oo and

_for

_any

function 0

with =

0(1)

_as

y ~ oo, there exist

W1,W2,... w8 which are

linearly independent

over

_Z,

and a

function f of

type 4D

such that

_{f (x)}

dx = 0 and

Theorem 21. Let s &#x3E;

_3,

and

_{f :}

T8 -+ R be smooth with zero mean

value. Assume that in the

_expansion

the

_coefficients

_{f,.,z1 ~", ~ma ,}

where

_{(rnl , ... ,}

₅₄

_0,

are all

different from

zero.

Then there exist _{WI, (û2, ... , £Us} which are

linearly independent

over Z such

that

An

_improvement

of the latter result was obtained

recently by

E.V.

Kolo-meikina

_[20].

One can see that the behaviour of

integrals

Jl

in two-dimensional case

radically

differs from the case s &#x3E; 3.

2.5. Odd functions.

_{Sergei Konyagin’s}

result.

_Recently,

S.

_Konyagin

[16]

obtained the

_following

result.

Theorem 22. The Kozlov’s

_conjecture

is true

_(that

is

_{(*) holds)}

_for

arbi-trary s &#x3E;

1 and _any

_{function f satisfying}

the condition

2.6. The smoothness. In

_[42],[41]

it is shown that we need some kind of

smoothness conditions on

_f

to insure that

_(*)

is true : indeed in the

two-dimensional case

(s

=

2),

there exists a function

f :

T2 --+

If8

(with

zero mean

value)

of the class such that

I(T, 0)

tends to

_infinity

when T - oo

(with

the choice _wi = 1 and _c.~2 =

v’2).

On the other

hand,

in

[44]

it is shown that when s =

2,

_a sufficient condition _on

f

for

having

(*),

is

f

to be

_absolutely

continuous.

Developping

an idea of D.V.

Treshchev,

the

author,

in

[31],

generalized

Poincare’s

_example.

He

_proved

that for _any real _{wl, ... ,} c~8 which form a

basis of a real

_algebraic

field,

there exists a function

f

E C’-2(T8)BCe-1(T’)

such that

_{1(., 0)}

does not

_satisfy

the

_property

_(*)

_{with cp}

= 0.

One may find some results on

_algebraic

numbers in

[40]

and

[38].

Re-cently,

S.V.

_Konyagin

_[16]

_proved

that for some Liouville transcendental

numbers,

there exists

_f

E with d ~

_28/s

such that

(*)

is not satis-fied.

Some

_early

results are reviewed in

[37].

2.7. Vector-functions:

_{counterexample}

in dimension s = 3. Let

f :

T’ -~ =

For _a~i,... ws E R be

_{linearly independent}

over

_Z,

we

_put

The

_analogue

of

_property

_(*)

for the

_{vector-integral}

I =

(Il, I2) :

If8 _-~

R2

becomes

Proposition

23. In the case when s = 2 and

f is

_a smooth

vector-func-tions,

then holds.

Proposition

24. The

_{analogue of}

Theorem 22 holds

_{for vector-function,}

that is

_(%)

is

_{satisfied for}

_any odd smooth

_{vector-function f .}

Let 4k : :

_{Il4 II8 F}

be a

_positive

function such that

L

4,(mM

’ " ’

’i

-"

Z

converges. A vector-function

f

=

( f 1, f 2) ;

T-1 -+

II82

is defined _to be

a

function of

_{type E}

if we have

Recently,

the author

_[34]

constructed the

_{following example.}

Theorem 25. For _any

_{given positive}

_function

_~,

there exist a

vector-func-tion

_f

=

( f 1,

f 2) ;

T3 -+

of

the zuith _{zero mean} value

=

0, j

=

1,

2)

and numbers _{c.y, W2 c.~3,} which are

_{linearly independent}

over Z

such that

Acknowledgment.

The author thanks

_prof.

M. Waldschmidt and

_prof.

P. Voutier for their

_help

in

_checking

the

_language

of the _paper.

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Nikolai G. MOSHCHEVITIN

Department of _Theory of Numbers

Faculty of Mathematics &#x26; mechanics Moscow State _University

Vorobiovy Gory

119899, Moscow, Russia E-mail : MOSBlnv.8&#x26;th.msu.su