inequality of logarithmic and asymptotic density

inequality of logarithmic and asymptotic density

For any

A

⊆

ℕ

we denote

A

⁢

(

n

)

:=

|

A

∩

{

1

,

2

,

…

,

n

}

|

and

S

⁢

(

n

)

:=

∑

k

=

1

n

1

k

.

Recall that the values

d

¯

⁢

(

A

)

=

lim inf

n

→

∞

⁡

A

⁢

(

n

)

n

d

¯

⁢

(

A

)

=

lim sup

n

→

∞

⁡

A

⁢

(

n

)

n

are called lower and

upper asymptotic density

of

A

.

The values

δ

¯

⁢

(

A

)

=

lim inf

n

→

∞

⁡

∑

k

∈

A

;

k

≤

n

1

k

S

⁢

(

n

)

δ

¯

⁢

(

A

)

=

lim sup

n

→

∞

⁡

∑

k

∈

A

;

k

≤

n

1

k

S

⁢

(

n

)

are called
lower and

upper logarithmic density

of

A

.

We have

S

⁢

(

n

)

∼

ln

⁡

n

(we use the

Landau notation

). This
follows from the fact that

lim

n

→

∞

⁡

S

⁢

(

n

)

-

ln

⁡

n

=

γ

is Euler’s
constant. Therefore we can use

ln

⁡

n

instead of

S

⁢

(

n

)

in the
definition of logarithmic density as well.

The sum in the definition of logarithmic density can be rewritten
using Iverson’s convention as

∑

k

=

1

n

1

k

[

k

∈

A

]

.
(This means that we only add elements fulfilling the condition

k

∈

A

. This notation is introduced in

[

1

, p.24]

.)

Theorem 1

.

For any subset

A

⊆

N

d

¯

⁢

(

A

)

≤

δ

¯

⁢

(

A

)

≤

δ

¯

⁢

(

A

)

≤

d

¯

⁢

(

A

)

holds.

Proof.

We first observe that

1

k

[

k

∈

A

]

=

A

⁢

(

k

)

-

A

⁢

(

k

-

1

)

k

,

D

(

n

)

:=

∑

k

=

1

n

1

k

[

k

∈

A

]

=

A

⁢

(

n

)

n

+

∑

k

=

1

n

-

1

A

⁢

(

k

)

k

⁢

(

k

+

1

)

There exists an

n

0

∈

ℕ

such that for each

n

≥

n

0

it holds

d

¯

⁢

(

A

)

-

ε

≤

A

⁢

(

n

)

n

≤

d

¯

⁢

(

A

)

+

ε

.

We denote

C

:=

1

+

S

⁢

(

n

0

)

. For

n

≥

n

0

we get

D

⁢

(

n

)

≤

C

+

∑

k

=

n

0

n

-

1

A

⁢

(

k

)

k

⋅

1

k

+

1

≤

C

+

(

d

¯

⁢

(

A

)

+

ε

)

⁢

∑

k

=

n

0

n

-

1

1

k

+

1

∼

(

d

¯

⁢

(

A

)

+

ε

)

⁢

ln

⁡

n

,

δ

¯

⁢

(

A

)

=

lim sup

n

→

∞

⁡

D

⁢

(

n

)

ln

⁡

n

≤

d

¯

⁢

(

A

)

+

ε

.

This

inequality

holds for any

ε

>

0

, thus

δ

¯

⁢

(

A

)

≤

d

¯

⁢

(

A

)

.

For the proof of the inequality for lower densities we put

C

′

:=

∑

k

=

1

n

0

-

1

A

⁢

(

k

)

k

⁢

(

k

+

1

)

-

(

d

¯

⁢

(

A

)

-

ε

)

⁢

S

⁢

(

n

0

)

.
We get

D

⁢

(

n

)

≥

C

′

+

(

d

¯

⁢

(

A

)

-

ε

)

⁢

S

⁢

(

n

0

)

+

(

d

¯

⁢

(

A

)

-

ε

)

⁢

∑

k

=

n

0

n

1

k

+

1

=

C

′

+

(

d

¯

⁢

(

A

)

-

ε

)

⁢

S

⁢

(

n

)

∼

(

d

¯

⁢

(

A

)

-

ε

)

⁢

ln

⁡

n

and this implies

δ

¯

⁢

(

A

)

≥

d

¯

⁢

(

A

)

.
∎

For the proof using Abel’s partial summation see

[

4

]

or

[

5

]

.

Corollary 1

.

If a set has asymptotic density, then it has logarithmic density,
too.

A well-known example of a set having logarithmic density but not
having asymptotic density is the set of all numbers with the first
digit equal to 1.

It can be moreover proved, that for any

real numbers

0

≤

α

¯

≤

β

¯

≤

β

¯

≤

α

¯

≤

1

there
exists a set

A

⊆

ℕ

such that

d

¯

⁢

(

A

)

=

α

¯

,

δ

¯

⁢

(

A

)

=

β

¯

,

δ

¯

⁢

(

A

)

=

β

¯

and

d

¯

⁢

(

A

)

=

α

¯

(see

[

2

]

).

References

1

R. L. Graham, D. E. Knuth, and O. Patashnik.

Concrete mathematics

. A foundation for computer science.

Addison-Wesley, 1989.

2

L. Mišík.

Sets of positive integers with prescribed values of densities.

Mathematica Slovaca

, 52(3):289–296, 2002.

3

H. H. Ostmann.

Additive

Zahlentheorie I

.

Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956.

4

J. Steuding.

http://www.math.uni-frankfurt.de/~steuding/steuding/prob.pdf

Probabilistic number theory.

5

G. Tenenbaum.

Introduction to

analytic

and probabilistic number theory

.

Cambridge Univ. Press, Cambridge, 1995.

Title

inequality of logarithmic and asymptotic density

Canonical name

InequalityOfLogarithmicAndAsymptoticDensity

Date of creation

2014-03-24 9:16:11

Last modified on

2014-03-24 9:16:11

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

8

Author

kompik (10588)

Entry type

Theorem

Classification

msc 11B05

Related topic

AsymptoticDensity

Related topic

LogarithmicDensity

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