# asymptotic density

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

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

 $\overline{d}(A)=\limsup_{n\rightarrow\infty}\frac{|A(n)|}{n}$

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

 $\underline{d}(A)=\liminf_{n\rightarrow\infty}\frac{|A(n)|}{n}$

We say $A$ has asymptotic density $d(A)$ if $\underline{d}(A)=\overline{d}(A)$, 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