Logarithmic density of a sequence of integers and density of its ratio set

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

L

ADISLAV

M

IŠÍK

J

ÁNOS

T. T

ÓTH

Logarithmic density of a sequence of integers

and density of its ratio set

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 309-318

<http://www.numdam.org/item?id=JTNB_2003__15_1_309_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

Logarithmic

density

of

a

_sequence

of

integers

and

_density

of

its ratio

set

par LADISLAV

MI0160ÍK

et

JÁNOS

T.

TÓTH

RÉSUMÉ. Nous donnons des conditions suffisantes pour que

l’en-semble

_R(A)

des fractions d’un ensemble d’entiers A soit dense dans

R+,

en termes des densités

_{logarithmiques}

de A. Ces

condi-tions different sensiblement de celles

_{précédemment}

obtenues en termes des densités

asymptotiques.

ABSTRACT. In the _paper sufficient conditions for the

_(R)-density

of a set of

_positive

_integers

in terms of

_logarithmic

densities are

given.

They

differ

_{substantially}

from those derived

previously

in

terms of

_asymptotic

densities.

1. Preliminaries

Denote

_by

N and the set of all

_{positive integers}

and

_positive

real

numbers,

respectively.

For A C N and x E let

_A(x)

=

f a

E

_A;

a

_~}.

Denote

_by

_{R(~4) ={~}

_{a E A,}

b E

_~4}

the ratio set

_of

A and _say that a

set A is

_(R)-dense

if

_R(A)

is

_{(topologically)}

dense in the set

_(see

_[3]).

Let us notice that the

(R)-density

of a set A is

_equivalent

to the

_density

of

R(A)

in the set

_{(1, oo).}

Define

. , i , . i . i ’II.

the lower

_{asymptotic density,}

_upper

_{asymptotic density,}

and

_asymptotic

density

(if defined),

respectively.

Similarly,

define

- , ... 1

the lower

_logarithmic

_density,

_upper

_{logarithmic density,}

and

_logarithmic

density

(if defined),

respectively.

Manuscrit reru le 3 decembre 2001.

The

_following

relations between

_asymptotic

_density

and

_(R)-density

are

known

(S2)

If

d(A) &#x3E; 2

then A is

(R)-dense

and for all b E

(0,

2 ~

there is a

(S2)

_{set B such that}

d(B)

= b and B is not

(R)-dense (see

[2], [5]

).

(S3)

If d(A)

= 1 then A is

(R)-dense

and for all b E

(0,1)

there is a

’ ’

set B such that

_d(B)

= b and B is not

(R)-dense (see

[3], [4] ).

Notice that the results

_{(Sl), (S2)}

and

_(S3)

can be formulated in a com-mon _way as results about maximal sets

_(with

_respect

to the

correspond-ing

density)

which are not

_(R)-dense

as follows. Denote

_by

D =

{~4

c

N;

A is not

_{(R) - dense}.}

Then we have

The aim of this _{paper is} to prove

corresponding

relations for

logarith-mic

_density.

_{It appears}

_(see

Theorem 2 and

_Corollary

₁₎

that

_they

differ

substantially

from the above ones for

_asymptotic

density.

2.

_Logarithmic

_density

and

_(R)-density

First,

let us introduce a useful

_technique

for calculation densities. It can be

_easily

seen that in

_practical

calculation of densities of a set

_A,

the

following

method can be used.

Write the set A as ...

are

_integers.

Then

and

In

_practice

the bounds _{pn, qn} of intervals

_determining

the set A are often

real numbers instead of

_integers.

Then it _may be convenient to use the

following

lemma. In

_fact,

we will use it in later calculations.

Lemma 1. Let 0 PI

ql ::;

p2

q2 ::; ... be

real numbers such

that oo and let

d,

Ibnl

d

for

each n E N and some

Proof.

For all n E N let _an,

_bn

be such that

2. both _{pn + an} and _{qn +}

_bn

are

_integers,

00

First,

let us notice that it is known that if for an

increasing

_sequence of

00

positive

integers

the series

_{E 1}

_{converges then} lim

_{Pn =}

0

_([1],

80.

n=l n n

Theorem,

p.124)

and

_trivially

also lim

_{n -}

0 for _any fixed r E II~.

n-+oo Pn r

Now a

_simple

analysis

shows that for each n E N

and, using

Lemma

_1,

As both lim and lim

_{pn d}

_{equal 0 an application}

of the

Sand-n

wich Theorem

_completes

the

_proof

for

_d(A).

In a _very similar _way one can

prove the

corresponding

statement for

_{d (A) .}

Let s be the first

_{positive integer}

i such that p2 - d &#x3E; 0. The

_following

inequalities hold for every

i = s, s -~- 1, ....

and

Again,

a

simple analysis

shows that

and,

using

(I)

and

_(II),

n

As the series

_2:

1

is

convergent

and lim 1 an

application

n-too

of the

Sandwich

Theorem

completes

the

_proof

_{for 6(A).}

In a _very similar

way one can _prove the

corresponding

statement for

6(A).

0

The

_{following simple}

lemma will be used in later calculations.

The

_following

class of sets

_plays

an

_important

role in our consideration.

where s =

minfn

E

_N;

-~-1

Theorem 1. Let 1 a b and A =

A(a, b)

_E A. Then

Proof.

(i)

Let x E

_{A, y E A}

and x _y.

_First,

let there be a n E N such

that both x

_{and y belong}

to the block

_(anbn

+

_1,

_an+1bn).

Then

On the other

_hand,

let

Then

In both

_{cases ~}

does not

_belong

to

_{(a, b) .}

(ii)

Calculate, using

Lemma

_1,

The

_{corresponding}

value of

_d(A)

can be calculated in a _very similar

(iii)

Again,

using

Lemma

_1,

we have

Remark 1. A

_{simple analysis}

of

_equalities

_(ii)

in Theorem 1 in

_comparison

to the results

_{(Sl), (S2), (S3)}

shows

A similar

_analysis

of

_equality

_(iii)

in Theorem 1 leads to the

_following.

Conjecture.

The

_following

_equalities

hold

The _purpose of the rest of _{this paper is} to _prove this

_conjecture.

All the

corresponding

results will be corollaries to the

_following.

Theorem 2. Let 1 a b and A =

A(a, b)

E A. Then the set A is rraaximal element in the set

_IX

C

_N‘;

_R(X)

n

_(a,

_b)

=

0}

with

_respect

_to the

partial

order induced

_by

_any

_3,

J.

Proof.

Let X C N be an infinite set such that

R(X)

n

_{(a, b)}

=

0.

Then X can be written in the form

are

_integers.

For the

_proof

it is sufficient to show

_(taking

into account Theorem 1

_(iii))

Thus we can also _suppose

The

_proof

will be carried in several

steps.

Step

1. In this

_step

we will _prove

and,

(a, b) = 0,

1,

qn)

a and

1+1

&#x3E; b.

p +1

&#x3E; b - a which

implies

a contradiction.

Step 2.

In this

_step

we will _prove

Proo, f of

(2).

Let Then also

_{and, by}

the

_previous

_step,

qn

+1)’

Suppose

on the

_contrary

that there exists x E X f1 +

Then

~1 = ~ ~

= b

_and,

as

R {X ) n

(~6)

==

0,

there exists m E n N such

_{that ji}

&#x3E; b and

_2013y

a.

Consequently

which

_implies

p. a contradiction.

Step

~. In this

_step

we will introduce some useful notation. Denote

by

I~o

the

smallest

_integer

I~ such that _Pk &#x3E;

_b

_{and let}

_{Ko =}

_{~o + 1,}

_{~o+2, ... }.}

From

₍₂₎

we have for

every k

E

_Ko

and so we can define a function

_Ko

-+

_Ko

_by

The _range of this function

_{~i

l ~

... )

is

_infinite,

denote

00

Kn =

for each n E N.

_Evidently

and for every

n=1

m n, x e

_Km

_{and y}

E

_Kn

it is x _y. Let us call a

big

_{gap in} X any

interval of the form +

₁₎

where k e

Kn

for n E N.

_Finally,

let us introduce two sequences

by

The above definitions

imply

that both

_{a(un + 1)}

and

_bvn

_belong

to the same

big

gap in X

and, consequently,

Step

.4.

In this

_step

we will

_present

and _prove

(if necessary)

some

simple

An easy

analysis proves

for all

_{positive integers}

_{p q.}

From Lemma 1 and

₍₀₎

it can be seen that

(7)

the series is

_divergent.

As

_a(ui+1

_{+ 1)}

_belongs

to the

_big

_gap next to the

_big

_gap in which

_{b(ui +1)}

lies we have _{ui+l + 1}

&#x3E; a (u~

+

₁₎

for every i E N. Therefore the series

1 is

convergent

and, consequently,

Ui

Step

5. For the rest of

_proof

_{suppose that} m is a

sufficiently

_large

fixed

positive integer

and denote

_{by n}

the

_greatest

_(fixed

from this

_moment)

positive integer

k for which

bvk

m.

_{Thus, using}

(5),

we have

The considerations in

_Step

3

_imply

that the intervals

_(pi

+ for i =

1, 2, ... ,

max

Kn

and

~a (u~ -~-1 ),

bvj)

for j

=

1, 2, ... ,

_{n are}

mutually

disjoint

and, by

definition of numbers m and _n,

_they

are all contained in the interval

~ l, m~ .

Thus, by

Lemma

_2,

we have

For similar reason we have

Step

7. In this

_step

we will

complete

the

proof by

limit _process. Let m -+ o0

(and

consequently

n e

oo).

_{Then, using}

(7)

and

_(8),

we have

The

_following

_corollary

is a direct _consequence of the

_previous

theorem

and it shows that the relations between

_(R)-density

and

_logarithmic

densi-ties are

_completely

different from those between

_(R)-density

and

_asymptotic

densities.

Corollary.

The

_following

relations hold

References

[1] K. KNOPP, Theory and _{Application of Infinite} Series. Blackie &#x26; Son _Limited, London and

Glasgow, 2-nd English Edition, 1957.

[2] O. STRAUCH, J. T. TÓTH, Asymptotic density of A C N and density of the ratio set R(A).

Acta Arith. 87 _(1998), 67-78. _Corrigendum in Acta Arith. 103 _(2002), 191-200.

[3] T.

_0160ALÁT,

On ratio sets _of sets _of natural numbers. Acta Arith. 15 _(1969), 173-278.

[4] T.

_0160ALÁT,

_{Quotientbasen} und _(R)-dichte mengen. Acta Arith. 19 (1971), 63-78.

[5] J. T. _TÓTH, Relation between _(R)-density and the lower _{asymptotic density.} Acta Math. Constantine the Philosopher University Nitra 3 _(1998), 39-44.

Ladislav Janos T. T6TH Department of Mathematics University of Ostrava 30. dubna 22 701 03 Ostrava Czech Republic