Upper 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:

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

Note that upper 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

• Lower asymptotic density
• Schnirelmann density