PlanetMath: asymptotic density

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asymptotic density

(Definition)

Let 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 by

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

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

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

We say

has asymptotic density

if $\underline{d}(A)=\overline{d}(A)$ , in which case we put $d(A)=\overline{d}(A)$ .

"asymptotic density" is owned by mathcam. [ full author list (2) ]

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Other names:

upper density, lower density, natural density, arithmetic density

Also defines:

upper asymptotic density, lower asymptotic density

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Cross-references: subset
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This is version 4 of asymptotic density, born on 2002-04-22, modified 2004-02-16.
Object id is 2861, canonical name is AsymptoticDensity.
Accessed 9303 times total.

Classification:

11B05 (Number theory :: Sequences and sets :: Density, gaps, topology)

Pending Errata and Addenda

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