Natural density

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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,

is 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 \mathbb{N}. 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 \mathbb{N}. Asymptotic density is also called arithmetic density.

Let A be a subset of the set of natural numbers \mathbb{N}=\{1,2,\ldots\}. For any n \in \mathbb{N} put A(n)=\{1,2,\ldots,n\} \cap A.

Define the upper asymptotic density \overline{d}(A) of A by

\overline{d}(A) = \limsup_{n \rightarrow \infty} \frac{| A(n)|}{n}

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

\underline{d}(A) = \liminf_{n \rightarrow \infty} \frac{ | A(n)| }{n}

We say A has asymptotic density d(A) if \underline{d}(A)=\overline{d}(A), in which case we put d(A)=\overline{d}(A).

This definition can be restated in the following way:

d(A)=\lim_{n \rightarrow \infty} \frac{| A(n)|}{n}

if the limit exists.

A somewhat weaker notion of density is upper Banach density; given a set A \subset \mathbb{N}, define d * (A) as

d^*(A) = \limsup_{N-M \rightarrow \infty} \frac{| A \bigcap \{M, M+1, ... , N\}|}{N-M+1}

[edit] References

  • M. Kolibiar, A. Legéň, T. Šalát and Š. Znám (1992). Algebra a príbuzné disciplíny. Alfa, Bratislava (in Slovak). ISBN 80-05-00721-3. 
  • H. H. Ostmann (1956). Additive Zahlentheorie I (in German). Berlin-Göttingen-Heidelberg: Springer-Verlag. 
  • G. Tenenbaum (1995). Introduction to analytic and probabilistic number theory. Cambridge: Cambridge Univ. Press. 


This article incorporates material from Asymptotic density on PlanetMath, which is licensed under the GFDL.

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