The $k$-dimensional Duffin and Schaeffer conjecture

A. D. P

OLLINGTON

R. C. V

AUGHAN

The k-dimensional Duffin and Schaeffer conjecture

Journal de Théorie des Nombres de Bordeaux, tome

1, n

o

_{1 (1989),}

p. 81-88

<http://www.numdam.org/item?id=JTNB_1989__1_1_81_0>

© Université Bordeaux 1, 1989, tous droits réservés.

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

The

k-dimensional

Duffin and Schaeffer

_conjecture.

par A.D. POLLINGTON AND R.C. VAUGHAN

Résumé - Nous montrons que la _conjecture de Duffin et Schaeffer est

vraie en toute dimension _supérieure à 1.

Abstract 2014 We show that the Duffin and Schaeffer _conjecture holds in all

dimensions greater than one.

In 1941 Duffin and Schaeffer

_[1]

made the

_following

_conjecture:

CONJECTURE. Let

_{an}

denote a _sequence of real numbers with

then the

_inequalities

have

_infinitely

_many solutions for almost aII x if and

_only

if

diverges.

If

₍₂₎

_converges then it

_easily

follows from the Borel-Cantelli lemma that the set of x’s

_satisfying

_infinitely

_many of the

_inequalities

₍₁₎

has

_Lebesgues

measure zero. Duffin and Schaeffer _gave conditions on the _an for which the

conjecture

is true and showed that the condition

_{(a, n)}

= 1 is _necessary. In

1970 Erd6s

_[2]

showed that the

_conjecture

holds if an

= 1

or 0. This was

later extended

_by

Vaaler

_[6]

who showed that an =

O( n )

is sufficient. In

his book on metric number

_theory

_[5]

_Sprindzuk

considers a k-dimensional

analogue

of the

_conjecture

in which

₍₁₎

is

_{replaced by}

where the measure is now

k-dimensional

Lebesgues

measure. He states that the

_study

of such

_{approximations}

_subject

to the conditions

_{(~i ,~) = "’ =}

(ak, n)

= 1 is

probably

a

_problem

of the same

_degree

of

_complexity

as the case n = 1. This

appears not to be the case. For we can now _prove the

k-dimensional

_analogue

of the Duflin and Schaeffer

_conjecture.

We _prove the

_following

result:

THEOREM. Let k &#x3E; 1 and let denote a _sequence of real numbers with

and _suppose that

diverges.

Then the

_inequalities

have

_infinitely

many solutions for almost all x E

Rk .

Unfortunately

our method does not

_readily

extend to the case k = 1. We

are able find some more _sequences for which the

_conjecture

_holds,

for

example

if an - 0 or 1 « an, but not to settle the dimension one case.

In the k-dimensional

_{case, k &#x3E;}

_2,

Vilchinski has

_previously

shown that we

may take an =

0(n-^f)

for

any I &#x3E; 0. Put

where

Thus

₍₄₎

becomes

diverges.

We are interested in

By

an

ergodic

theorem of

Gallagher

[3],

see

Sprindzuk

[5],

this is either

zero or one.

Gallagher

_proves his result for dimension one. The

correspon-ding

k-dimensional result is

_{proved by}

Vilchinski

_[7].

In order to _prove the theorem it therefore suffices to show that

₍₈₎

is not zero.

Since

₍₇₎

_diverges,

and

_{.Bk(En) -~}

0 as n - _oo,

given

_{any 1} &#x3E; _1/ &#x3E;

_0,

for

every N we can find a finite set Z so that if z E Z then z &#x3E;

_N,

and

By

the

_{Cauchy-Schwarz inequality}

this is Lemma 5 of

_Sprindzuk

_[5].

So

_provided

there is some absolute

Thus we need to bound

In the

_subsequent

discussion constants in the

_{Vinogradov « symbol}

de-pend

at most on the dimension k.

Note that

and

We now concentrate on k = 1 and _use

(12)

and

(13)

to obtain a bound for

k&#x3E;1.

We wish to obtain an estimate for

Put

and

Then

By

estimating

LEMMA 1. With

En

as above and d =

(m, n)

there is an absolute constant

c so that

Let

_g(n)

be defined as the least number so that

Put

Then

_by

Lemma 1

In

_particular

if

_{2= &#x3E; t}

then

Now

We now return to the k-dimensional case.

_By

(12)

and

(13)

Recall that m n so

(c)

is

impossible.

We will consider the other three

cases

seperately.

The first case

_corresponds

to the situation considered

_by

Erdos .

Case

_(a).

We have A

We need to consider

_pairs

_{m, n} with

Let denote that

_part

of the sum

for which

_g(m’)

= u and

g(n’)

= _v.

Then, by

Lemma 1 and

(22)

Where

_L(u)

means the sum of those m’ with

g(m’) = u

and m = m’d.

then

To estimate the inner sum we will use the

following

result due to Erd6s which also _appears in Vaaler

_[6].

Let

_N(~,

_v,

_x)

be the number of

_integers

n x which

_satisfy

-Then

LEMMA 2. For _any E &#x3E; 0 and

_{any ~}

&#x3E; 0 there exists a

_positive

_integer

vo =

vo(~, c)

such that for all x _&#x3E; 0 and all _{v &#x3E;}

vo,

where

_{log /3}

=

~.

Applying

this result

_{with ~}

= 1 and

,C~(1 - c)

= 2

we have

COROLLARY. Given x &#x3E; 0 and v &#x3E; 1

Using

(23)

and

_partial

summation

Applying

(23)

again

The other cases are similar and we have

where c is a constant

_{depending only}

on k. This

_completes

the

_proof

of the

Theorem .

REFERENCES

1. R.J. Duffin and A.C. _Schaeffer, Khintchine’s _problem in metric _Diophantine

ap-proximation, Duke Math. J. 8 _(1941), 243-255.

2. _P.Erdös, On the distribution _{of convergents of} almost all real _numbers, J. Number

Theory 2 _(1970), 425-441.

3. P.X. _{Gallagher, Approximation by} reduced fractions, J. Math. Soc. of Japan 13

(1961), 342-345.

4. Halberstam and _Richert, "Sieve _methods," Academic _{Press, London,} 1974. 5. V.G. _Sprindzuk, "Metric theory of Diophantine approximations," V.H. Winston

and _{Sons, Washington D.C.,} 1979.

6. J.D. _Vaaler, On the metric _{theory of Diophantine approximation,} Pacific J. Math.

76 _(1978), 527-539.

7. V.T. _Vilchinski, On simultaneous _{approximations,} Vesti Akad Navuk BSSR Ser

Fiz.-Mat _(1981), 41-47.

8. , The Duffin and

Schaeffer

conjecture and simultaneous

approxi-mations, Dokl. Akad. Nauk BSSR 25 _(1981), 780-783.

Mots clefs: Diophantine approximation, k- _{dimensional, Lebesgues measure,} Duffin and Schaeffer _conjecture.

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