NationMaster - Encyclopedia: Natural density

This article may be too technical for a general audience.
Please help improve this article by providing more context and better explanations of technical details to make it more accessible, without removing technical details.

This template is misplaced. It belongs on the talk page: Talk:Natural density.

In number theory, asymptotic density or natural density is one of the possibilities to measure how large is a subset of the set of natural numbers $mathbb{N}$ . Image File history File links Information. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, the term small set is sometimes used to refer to any set that is small, a subjective concept. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

If we pick randomly a number from the set ${1,2,ldots,n}$ , then the probability that it belongs to A is the ratio of the number of elements of sets $Acap{1,2,ldots,n}$ and $A$ . If this probability tends to some limit as n tends to infinity, then we call this limit the asymptotic density of A. We see that this notion can be understood as a kind of probability of chosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in the probabilistic number theory. Probabilistic number theory is a subfield of number theory, which uses explicitly probability to answer questions of number theory. ...

Asymptotic density contrasts, for example, with the Schnirelmann density. A drawback of this approach is that the asymptotic density is not defined for all subsets of $mathbb{N}$ . Asymptotic density is also called arithmetic density. In mathematics, the Schnirelmann density of a sequence of numbers is a way to measure how dense the sequence is. ...

1 Definition
- 1.1 Upper and lower asymptotic density
2 Examples
3 References

Definition

A sequence

a₁, a₂, ... , a_n, .....

with the a_j positive integers and Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

a_j < a_j+1 for all j,

has natural density (or asymptotic density) α, where

0 ≤ α ≤ 1,

if the proportion of natural numbers included as some a_j is asymptotic to α. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

More formally, if we define the counting function A(x) as the number of a_j's with

a_j < x

then we require that

A(x) ~ αx as x → +∞.

Upper and lower asymptotic density

Let $A$ be a subset of the set of natural numbers $mathbb{N}={1,2,ldots}.$ For any $n in mathbb{N}$ put $A(n)={1,2,ldots,n} cap A.$

Define the upper asymptotic density $overline{d}(A)$ of $A$ by

$overline{d}(A) = limsup_{n rightarrow infty} frac{| A(n)|}{n}$

$overline{d}(A)$ is also known simply as the upper density of $A .$ Similarly, we define $underline{d}(A)$ , the lower asymptotic density of $A$ , by

$underline{d}(A) = liminf_{n rightarrow infty} frac{ | A(n)| }{n}$

We say $A$ has asymptotic density $d (A)$ if $underline{d}(A)=overline{d}(A)$ , in which case we put $d(A)=overline{d}(A).$

This definition can be restated in the following way:

$d(A)=lim_{n rightarrow infty} frac{| A(n)|}{n}$

if the limit exists.

A somewhat weaker notion of density is upper Banach density; given a set $A subset mathbb{N}$ , define $d * (A)$ as

$d^*(A) = limsup_{N-M rightarrow infty} frac{| A bigcap {M, M+1, ... , N}|}{N-M+1}$

If we write a subset of $mathbb{N}$ as an increasing sequence

$A={a_1<a_2<ldots<a_n<ldots; ninmathbb{N}}$

then

$underline{d}(A) = liminf_{n rightarrow infty} frac{n}{a_n},$

$overline{d}(A) = liminf_{n rightarrow infty} frac{n}{a_n}$

and $d(A) = lim_{n rightarrow infty} frac{n}{a_n}$ if the limit exists.

Examples

For any finite sets F of positive integers d(F)=0.

If $A={n^2; ninmathbb{N}}$ is the set of all squares, then 'd(A)=0'.

If $A={2n; ninmathbb{N}}$ is the set of all even numbers, then $d(A)=frac12$ .

For the set P of all primes we get from Prime number theorem d(P)=0. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

The set $A=bigcuplimits_{n=0}^infty {3^{2n},ldots,3^{2n+1}-1}$ is an example of a set which does not have asymptotic density, since the upper density of this set is $overline d(A)=frac 23$ and the lower density is $underline d(A)=frac 13$ .

References

M. Kolibiar, A. Legéň, T. Šalát and Š. Znám (1992). Algebra a príbuzné disciplíny. Alfa, Bratislava (in Slovak). ISBN 80-05-00721-3.

H. H. Ostmann (1956). Additive Zahlentheorie I (in German). Berlin-Göttingen-Heidelberg: Springer-Verlag.

Steuding, Jörn. Probabilistic number theory. Retrieved on 2005-10-06.

G. Tenenbaum (1995). Introduction to analytic and probabilistic number theory. Cambridge: Cambridge Univ. Press.

This article incorporates material from Asymptotic density on PlanetMath, which is licensed under the GFDL. 2005 (MMV) was a common year starting on Saturday of the Gregorian calendar. ... October 6 is the 279th day of the year (280th in leap years). ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

Categories: Wikipedia articles that are too technical | Main pages with misplaced talk page templates | Number theory | Combinatorics

Results from FactBites:

Energy Density of Natural Gas (434 words)
	Natural gas, a combustible mixture of hydrocarbons, is a very important source of energy since it is clean, cheap and efficient.
	The first is that natural gas is formed when organic matter, such as the remains of a plant or animal, is compressed beneath the earth at high pressures for a long period of time.
	For natural gases, the energy density is the either the amount of energy stored per unit volume or per unit mass of the gas.

All the chemical fusion reactions release energy and all the chemical fission reactions absorb energy (926 words)
	Natural density of atomic shells: Densities of space matter in the shells of an atom in constant temperature and pressure.
	The greater space matter density is in the most inner K shell and gradually decreases to the outer shells (more details about the structure of an atom and evidences for the space matter in the atomic shells are provided in my article "Structure of an atom".
	Natural densities of atomic shells of hydrogen and oxygen (shells that participated in the reaction)

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Definition GA_googleFillSlot("encyclopedia_square");

Upper and lower asymptotic density

Examples

References

Definition