asymptotic density

Let $A$ be a subset of ${\mathbb{Z}}^{+}$. For any $n\in {\mathbb{Z}}^{+}$ put $A(n)=\{1,2,\mathrm{\dots },n\}\cap A$.

Define the upper asymptotic density $\overline{d}(A)$ of $A$ by

$$\overline{d}(A)=\underset{n\to \mathrm{\infty }}{lim\; sup}\frac{|A(n)|}{n}$$

$\overline{d}(A)$ is also known simply as the upper density of $A$.
Similarly, we define $\underset{¯}{d}(A)$, the lower asymptotic density of $A$, by

$$\underset{¯}{d}(A)=\underset{n\to \mathrm{\infty }}{lim\; inf}\frac{|A(n)|}{n}$$

We say $A$ has asymptotic density $d(A)$ if $\underset{¯}{d}(A)=\overline{d}(A)$, in which case we put $d(A)=\overline{d}(A)$.

Title	asymptotic density
Canonical name	AsymptoticDensity
Date of creation	2013-03-22 12:36:20
Last modified on	2013-03-22 12:36:20
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 11B05
Synonym	upper density
Synonym	lower density
Synonym	natural density
Synonym	arithmetic density
Related topic	InequalityOfLogarithmicAndAsymptoticDensity
Defines	upper asymptotic density
Defines	lower asymptotic density