Difference between revisions of "Asymptotic density"
(TeX) |
(→References: isbn link) |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{TEX|done}} | {{TEX|done}} | ||
| − | 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 |
$$ \alpha=\liminf_{x\to\infty}\frac{A(x)}{x},$$ | $$ \alpha=\liminf_{x\to\infty}\frac{A(x)}{x},$$ | ||
| Line 12: | Line 12: | ||
$$\beta=\limsup_{x\to\infty}\frac{A(x)}{x}$$ | $$\beta=\limsup_{x\to\infty}\frac{A(x)}{x}$$ | ||
| − | 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]]. |
| − | |||
| − | |||
====References==== | ====References==== | ||
{| | {| | ||
|- | |- | ||
| − | |valign="top"|{{Ref|HaRo}}||valign="top"| H. Halberstam, K.F. Roth, "Sequences" , '''1''' , Clarendon Press (1966) | + | |valign="top"|{{Ref|HaRo}}||valign="top"| H. Halberstam, K.F. Roth, "Sequences" , '''1''' , Clarendon Press (1966) {{ZBL|0141.04405}} (repr. 1983) {{ISBN|0-387-90801-3}} {{ZBL|0498.10001}} |
|- | |- | ||
|} | |} | ||
Latest revision as of 13:38, 25 November 2023
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=25046