Asymptotic density
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) Zbl 0141.04405 (repr. 1983) ISBN 0-387-90801-3 Zbl 0498.10001 |
Asymptotic density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_density&oldid=37571