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A variant of the general concept of the [[Density of a sequence|density of a sequence]] of natural numbers; which measures how large a part of the sequence of all natural numbers belongs to the given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136301.png" /> of natural numbers including zero. The asymptotic density of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136302.png" /> is expressed by the real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136303.png" /> defined by the formula
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A variant of the general concept of the [[Density of a sequence|density of a sequence]] of natural numbers; which measures how large a part of the sequence of all natural numbers belongs to the given sequence $A$ of natural numbers including zero. The asymptotic density of a sequence $A$ is expressed by the real number $\alpha$ defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136304.png" /></td> </tr></table>
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$$ \alpha=\liminf_{x\to\infty}\frac{A(x)}{x},$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136305.png" /></td> </tr></table>
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$$ A(x)=\sum_{\substack{a\in A\\0<a\leq x}}1,\quad x\geq 1.$$
  
 
The number
 
The number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136306.png" /></td> </tr></table>
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$$\beta=\limsup_{x\to\infty}\frac{A(x)}{x}$$
  
is known as the upper asymptotic density. If the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136308.png" /> coincide, their common value is called the natural density. Thus, for instance, the sequence of numbers that are free from squares has the natural density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a0136309.png" />. The concept of an asymptotic density is employed in finding criteria for some sequence to be an [[Asymptotic basis|asymptotic basis]].
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is known as the upper asymptotic density. If the numbers $\alpha$ and $\beta$ coincide, their common value is called the natural density. Thus, for instance, the sequence of numbers that are free from squares has the natural density $\delta=6/\pi^2$. The concept of an asymptotic density is employed in finding criteria for some sequence to be an [[Asymptotic basis|asymptotic basis]].
  
 
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The number $\alpha$ as defined above is also called the lower asymptotic density.
 
 
====Comments====
 
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013630/a01363010.png" /> as defined above is also called the lower asymptotic density.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Halberstam,  K.F. Roth,  "Sequences" , '''1''' , Clarendon Press  (1966)</TD></TR></table>
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|valign="top"|{{Ref|HaRo}}||valign="top"| H. Halberstam,  K.F. Roth,  "Sequences" , '''1''' , Clarendon Press  (1966)
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Revision as of 12:59, 22 April 2012

A variant of the general concept of the density of a sequence of natural numbers; which measures how large a part of the sequence of all natural numbers belongs to the given sequence $A$ of natural numbers including zero. The asymptotic density of a sequence $A$ is expressed by the real number $\alpha$ defined by the formula

$$ \alpha=\liminf_{x\to\infty}\frac{A(x)}{x},$$

where

$$ A(x)=\sum_{\substack{a\in A\\0<a\leq x}}1,\quad x\geq 1.$$

The number

$$\beta=\limsup_{x\to\infty}\frac{A(x)}{x}$$

is known as the upper asymptotic density. If the numbers $\alpha$ and $\beta$ coincide, their common value is called the natural density. Thus, for instance, the sequence of numbers that are free from squares has the natural density $\delta=6/\pi^2$. The concept of an asymptotic density is employed in finding criteria for some sequence to be an asymptotic basis.

The number $\alpha$ as defined above is also called the lower asymptotic density.

References

[HaRo] H. Halberstam, K.F. Roth, "Sequences" , 1 , Clarendon Press (1966)
How to Cite This Entry:
Asymptotic density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_density&oldid=12011
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article