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 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)