In number theory, asymptotic density or natural density is one of the possibilities to measure how large is a subset of the set of natural numbers <math>\mathbb{N}</math>.
Intuitively, we feel that there are “more” odd numbers than perfect squares; however, the set of odd numbers is not in fact “bigger” than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-one correspondence. Clearly, we need a better way to formalize our intuitive notion.
If we pick randomly a number from the set <math>\{1,2,\ldots,n\}</math>, then the probability that it belongs to A is the ratio of the number of elements in the set <math>A\cap\{1,2,\ldots,n\}</math> and n. If this probability tends to some limit as n tends to infinity, then we call this limit the asymptotic density of A. We see that this notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in the probabilistic number theory.
Asymptotic density contrasts, for example, with the Schnirelmann density.
A drawback of this approach is that the asymptotic density is not defined for all subsets of <math>\mathbb{N}</math>. Asymptotic density is also called arithmetic density.
Definition
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 → +∞.
Upper and lower asymptotic density
Let <math>A</math> be a subset of the set of natural numbers <math>\mathbb{N}=\{1,2,\ldots\}.</math> For any <math>n \in \mathbb{N}</math> put <math>A(n)=\{1,2,\ldots,n\} \cap A.</math>
Define the upper asymptotic density <math>\overline{d}(A)</math> of <math>A</math> by
- <math> \overline{d}(A) = \limsup_{n \rightarrow \infty} \frac{| A(n)|}{n} </math>
<math>\overline{d}(A)</math> is also known simply as the upper density of <math>A.</math> Similarly, we define <math>\underline{d}(A)</math>, the lower asymptotic density of <math>A</math>, by
- <math> \underline{d}(A) = \liminf_{n \rightarrow \infty} \frac{ | A(n)| }{n} </math>
We say <math>A</math> has asymptotic density <math>d(A)</math> if <math>\underline{d}(A)=\overline{d}(A)</math>, in which case we put <math>d(A)=\overline{d}(A).</math>
This definition can be restated in the following way:
- <math> d(A)=\lim_{n \rightarrow \infty} \frac{| A(n)|}{n} </math>
if the limit exists.
A somewhat weaker notion of density is upper Banach density; given a set <math>A \subset \mathbb{N}</math>, define <math>d^*(A)</math> as
- <math> d^*(A) = \limsup_{N-M \rightarrow \infty} \frac{| A \bigcap \{M, M+1, … , N\}|}{N-M+1} </math>
If we write a subset of <math>\mathbb{N}</math> as an increasing sequence
- <math> A=\{a_1<a_2<\ldots<a_n<\ldots; n\in\mathbb{N}\}</math>
then
- <math>\underline{d}(A) = \liminf_{n \rightarrow \infty} \frac{n}{a_n},</math>
- <math>\overline{d}(A) = \limsup_{n \rightarrow \infty} \frac{n}{a_n}</math>
and
<math>d(A) = \lim_{n \rightarrow \infty} \frac{n}{a_n}</math>
if the limit exists.
Examples
Obviously, d(N) = 1.
For any finite set F of positive integers, d(F) = 0.
If <math>A=\{n^2; n\in\mathbb{N}\}</math> is the set of all squares, then d(A) = 0.
If <math>A=\{2n; n\in\mathbb{N}\}</math> is the set of all even numbers, then d(A) = 1/2. Similarly, for any arithmetical progression <math>A=\{an+b; n\in\mathbb{N}\}</math> we get d(A) = 1/a.
For the set P of all primes we get from the prime number theorem d(P) = 0.
The set
<math>A=\bigcup\limits_{n=0}^\infty \{3^{2n},\ldots,3^{2n+1}-1\}</math>
is an example of a set which does not have asymptotic density, since the upper density of this set is
<math>\overline d(A)=\frac 23</math>
and the lower density is
<math>\underline d(A)=\frac 13</math>.
References
- Document Scanner Software | Captiva Software - QuickScan Pro Out-of-the-box document scanner software solution that provides all the necessary capabilities for high-speed scanning, image enhancement, OCR, viewing,
- Document Management Services at WorldView Ltd.: Document Document Management Services at WorldView Ltd.. Specializing in document management software, enterprise content management, document management systems and
- Welcome to ScanStore - your source for document imaging, document We specialize in low-cost, easy-to-implement document scanning, document management, OCR and forms processing software for small and medium sized businesses
- Document Management Document Imaging Document System - Laserfiche® Document imaging and document management software for records management professionals. Manages, stores and retrieves millions of pages across the
- Vixelsoft - Document Imaging Software - Document Management and Your paperless office document management software solution for small to medium business and home user.
- Document Imaging Solutions and Scanning Software The Document Imaging Solutions Briefing Center provides the information and insight necessary for IT professionals to get the most from document imaging
- UM System Wide VPA for Document Imaging Software with Perceptive Scanner hardware for use with ImageNow is also available from Perceptive Software, Inc. Specific model choices should be made after initial planning and
- OCR Software - OCR Optical Character Recognition by ReadSoft But document automation means so much more than just OCR software. The OCR software part deals mainly with document scanning, document capture and data
- Document Scanning and Imaging Software This variant can also be used for single document scanning at home only - please do not use this for business purposes. Please note that batch scanning and
- Electronic Document Management Software: document imaging software document imaging software specifically for the SMB marketplace. VersaIMAGE-GOLD offers scanning, auto indexing with MS database image management and
- HP Document Management Solutions - document scanning, management HP's document and content management solutions include document scanning appliances, document processing and routing software, servers, storage,
- Q&A with Tim Captain, Associate Editor, BLI How important is it for document scanning software to integrate seamlessly with other business applications like content management, email and collaboration
- SimpleIndex: the best value in document scanning and automated The SimpleIndex Document Imaging Suite consists of five products that combine to form a Keywords: document imaging, imaging software, scanning software,
- Perceptive Software and ImageNow Document Imaging, Document Perceptive Software develops ImageNow document imaging, document management and workflow (ECM) solutions used by organizations worldwide in a broad range of