Natural density
From Wikipedia, the free encyclopedia
In mathematics, a sequence
- a1, a2, ... , an,
with the aj positive integers and
- aj < aj+1 for all j,
has natural density (or asymptotic density) α, where
- 0 ≤ α ≤ 1,
if the proportion of natural numbers included as some aj is asymptotic to α.
More formally, if we define the counting function A(x) as the number of aj's with
- aj < x
then we require that
- A(x) ~ αx as x → +∞.
[edit] Formal definitions
The asymptotic density is one way to measure how large is a subset of the set of natural numbers 
. It contrasts, for example, with the Schnirelmann density. A drawback of this approach is that the asymptotic density is not defined for all subsets of . Asymptotic density is also called arithmetic density.
Let A be a subset of the set of natural numbers 
For any 
put 
Define the upper asymptotic density 
of A by
is also known simply as the upper density of A. Similarly, we define 
, the lower asymptotic density of A, by
We say A has asymptotic density d(A) if 
, in which case we put 
This definition can be restated in the following way:
if the limit exists.
A somewhat weaker notion of density is upper Banach density; given a set 
, define d * (A) as
[edit] References
This article incorporates material from Asymptotic density on PlanetMath, which is licensed under the GFDL.
