Difference between revisions of "Asymptotic density"
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| − | A variant of the general concept of the [[ | + | 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 ''(lower) asymptotic density'' of a sequence $A$ is expressed by the real number $\alpha$ defined by the formula |
$$ \alpha=\liminf_{x\to\infty}\frac{A(x)}{x},$$ | $$ \alpha=\liminf_{x\to\infty}\frac{A(x)}{x},$$ | ||
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$$\beta=\limsup_{x\to\infty}\frac{A(x)}{x}$$ | $$\beta=\limsup_{x\to\infty}\frac{A(x)}{x}$$ | ||
| − | 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 [[ | + | 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]]. |
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====References==== | ====References==== | ||
Revision as of 21:12, 16 January 2016
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 (lower) 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.
References
| [HaRo] | H. Halberstam, K.F. Roth, "Sequences" , 1 , Clarendon Press (1966) |
Asymptotic density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_density&oldid=25046