Natural density

Remarks

The natural density exists if and only if $\overline{d}(A) = \underline{d}(A)$, and then all three numbers are equal.

Example: for the multiples of a positive integer $m$, $A(n) = \floor{n/m}$, and $n/m - 1 < A(n) \le n/m$; so $d(A) = 1/m$.

Non-example: let $A$ be the set of positive integers whose first decimal digit is $1$. For $n = 2 \cdot 10^k - 1$ one has $A(n) = 1 + 10 + \dots + 10^k = (10^{k+1} - 1)/9$, so $A(n)/n \to 5/9$. For $n = 10^{k+1} - 1$ the count $A(n)$ is the same, so $A(n)/n \to 1/9$. Hence the limit does not exist; here $\overline{d}(A) = 5/9$ and $\underline{d}(A) = 1/9$.

If $d(A)$ and $d(B)$ exist and $A \cap B$ is empty, then $d(A \cup B) = d(A) + d(B)$. Natural density is not the same as Schnirelmann density, which is an infimum over all $n$ and is $0$ for every set not containing $1$.