Vipul: Created page with '{{density notion}} ==Definition== Let $S$ be a subset of the set $\mathbb{N}$ of natural numbers. The '''asymptotic density''' or '''natural density''' of...'

2009-05-06T21:27:03Z

Created page with '{{density notion}} ==Definition== Let <math>S</math> be a subset of the set <math>\mathbb{N}</math> of natural numbers. The '''asymptotic density''' or '''natural density''' of...'

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{{density notion}}

==Definition==

Let <math>S</math> be a subset of the set <math>\mathbb{N}</math> of natural numbers. The '''asymptotic density''' or '''natural density''' of <math>S</math> is defined as the following quantity, if well-defined:

<math>\lim_{n \to \infty} \frac{S \cap \{ 1,2, \dots, n \}}{n}</math>.

The asymptotic density need not be well-defined. However, the ''limit superior'' and ''limit inferior'' of the above sequence are always well-defined, and these are termed respectively the [[upper asymptotic density]] and [[lower asymptotic density]].

==Related notions==

* [[Lower asymptotic density]]
* [[Upper asymptotic density]]
* [[Schnirelmann density]]

Asymptotic density - Revision history

Vipul: Created page with '{{density notion}} ==Definition== Let S be a subset of the set \mathbb{N} of natural numbers. The '''asymptotic density''' or '''natural density''' of...'

Vipul: Created page with '{{density notion}} ==Definition== Let $S$ be a subset of the set $\mathbb{N}$ of natural numbers. The '''asymptotic density''' or '''natural density''' of...'