asymptotic density

Let A be a subset of +. For any n+ put A(n)={1,2,,n}A.

Define the upper asymptotic density d¯(A) of A by

d¯(A)=lim supn|A(n)|n

d¯(A) is also known simply as the upper density of A.
Similarly, we define d¯(A), the lower asymptotic density of A, by

d¯(A)=lim infn|A(n)|n

We say A has asymptotic density d(A) if d¯(A)=d¯(A), in which case we put d(A)=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