Lower asymptotic density

Definition

Let $S$ be a subset of the set $\mathbb{N}$ of natural numbers. The upper asymptotic density of $S$ in $\mathbb{N}$ is defined as:

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

Note that lower asymptotic density is unchanged upon the addition or removal of a finite subset. In particular, the upper asymptotic density is the same both as a subset of $\mathbb{N}$ and as a subset of ${\mathbb{N}}_0$ (the natural numbers along with zero).

Related notions

• Upper asymptotic density
• Schnirelmann density