Asymptotic density

Definition

Let $S$ be a subset of the set $\mathbb{N}$ of natural numbers. The asymptotic density or natural density of $S$ is defined as the following quantity, if well-defined:

$\underset{n\to \mathrm{\infty }}{lim}\frac{S\cap \{1,2,\dots ,n\}}{n}$.

The asymptotic density need not be well-defined. However, the limit superior and limit inferior of the above sequence are always well-defined, and these are termed respectively the upper asymptotic density and lower asymptotic density.

Related notions

• Lower asymptotic density
• Upper asymptotic density
• Schnirelmann density