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Natural Density

The natural density (or asymptotic density) of a set A of positive integers, if it exists, is

d(A)=lim_(n->infty)(|A intersection {1,...,n}|)/n.

The lower and upper natural densities are obtained by replacing the limit with the infimum limit and supremum limit, respectively. The natural density exists exactly when these two quantities are equal (Nathanson and Parikh 2007, p. 152).

In the terminology of dynamical systems, natural density can instead mean the natural invariant, also called the invariant density.

See also

Banach Density, Logarithmic Density, Lower Banach Density, Natural Invariant, Schnirelmann Density, Sequence Density, Upper Banach Density

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References

Nathanson, M. B. and Parikh, R. "Density of Sets of Natural Numbers and the Lévy Group." J. Number Th. 124, 151-158, 2007. https://doi.org/10.1016/j.jnt.2006.09.001.

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Natural Density

Cite this as:

Weisstein, Eric W. "Natural Density." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NaturalDensity.html

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