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

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.

Full text

JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
LADISLAV MIŠÍK
JÁNOS T. TÓTH
Logarithmic density of a sequence of integers
and density of its ratio set
Journal de Théorie des Nombres de Bordeaux, tome 15, no1 (2003),
p. 309-318
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309
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
research
was
supported
by
Grant
GA
CR
201/01/0471.

310
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
n=1

311
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,
or,
rewritten,

312
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.
Lemma
2.
Let
and
such
that
1
=

313
Proof.
The
statement
of
the
Lemma
is
a
straightforward
consequence
of
the
following
relations
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
way.

314
(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

315
Proof
of
(1).
S u ose &#x3E;
+
1).
Then
a
and
also
9
a
f
f
( )
PP
’n
_
p
+
1 .
Then
P,., +1 -
P,+l
and,
as
rl
(a,
b) =
0,
there
exists
m
E
+
1,
qn)
n
N
such
that
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
relations
and
statement
which
will
be
used
in
the
final
calculation.

316
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
The
last
inequality
together
with
(5)
imply

317
Step
6.
Denote
c
.
Now
we
are
able
to
estimate
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.

318
Corollary.
The
following
relations
hold
References
[1]
K.
KNOPP,
Theory
and
Application
of
Infinite
Series.
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Son
Limited,
London
and
Glasgow,
2-nd
English
Edition,
1957.
[2]
O.
STRAUCH,
J.
T.
TÓTH,
Asymptotic
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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.
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Arith.
15
(1969),
173-278.
[4]
T.
0160ALÁT,
Quotientbasen
und
(R)-dichte
mengen.
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Arith.
19
(1971),
63-78.
[5]
J.
T.
TÓTH,
Relation
between
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and
the
lower
asymptotic
density.
Acta
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Nitra
3
(1998),
39-44.
Ladislav
Janos
T.
T6TH
Department
of
Mathematics
University
of
Ostrava
30.
dubna
22
701
03
Ostrava
Czech
Republic
E-mail :
ladislav.misik4losu.cz,
tothoosu.cz