Natural Density

a with the ''a''_''j'' Positive Integer s and a has natural density (or '''asymptotic density''') α, where :0 ≤ α ≤ 1, if the proportion of Natural Number s included as some ''a''_''j'' is Asymptotic to α. More formally, if we define the counting function ''A''(''x'') as the number of ''a''_''j'' with a then we require that A 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
	:<math> \underline{d}(A)	\liminf_{n ightarrow \infty} rac{ A(n) }{n} </math>

	CATEGORIES ABOUT NATURAL DENSITY

	number theory

	combinatorics

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