Upper asymptotic density: Difference between revisions
(Created page with '{{density notion}} ==Definition== Let <math>S</math> be a subset of the set <math>\mathbb{N}</math> of natural numbers. The '''upper asymptotic density''' of <math>S</math> in ...') |
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<math>\lim \sup_{n \to \infty} \frac{S \cap \{ 1,2, \dots, n \} }{n}</math>. | <math>\lim \sup_{n \to \infty} \frac{S \cap \{ 1,2, \dots, n \} }{n}</math>. | ||
Note that upper asymptotic density is unchanged upon the addition or removal of a finite subset. In particular, the upper asymptotic density is the same both as a subset of <math>\mathbb{N}</math> and as a subset of <math>\mathbb{N}_0</math> (the natural numbers along with zero). | |||
==Related notions== | ==Related notions== | ||
Latest revision as of 21:21, 6 May 2009
Definition
Let be a subset of the set of natural numbers. The upper asymptotic density of in is defined as:
.
Note that upper asymptotic density is unchanged upon the addition or removal of a finite subset. In particular, the upper asymptotic density is the same both as a subset of and as a subset of (the natural numbers along with zero).
