asymptotic densityAsymptoticDensity2002-04-22 13:27:302004-02-16 15:36:26Definitionupper asymptotic densitylower asymptotic density% this is the default PlanetMath preamble. as your knowledge
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% there are many more packages, add them here as you need themLet $A$ be a subset of $\mathbb{Z}^{+}$. For any $n \in \mathbb{Z}^{+}$ put
$A(n)=\{1,2,\ldots,n\} \cap A$.
Define the {\em 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 {\em upper density} of
$A$.\\
Similarly, we define $\underline{d}(A)$, the {\em lower asymptotic density} of
$A$, by
\[ \underline{d}(A) = \liminf_{n \rightarrow \infty} \frac{ |A(n)| }{n} \]
We say $A$ has {\em asymptotic density} $d(A)$ if
$\underline{d}(A)=\overline{d}(A)$, in which case we put
$d(A)=\overline{d}(A)$.