Let $A$ be a subset of $\mathbb{Z}$ and let $A(n)$ be number of elements of $A$ in $[1,n]$ Schnirelmann density of $A$ is \begin{equation*} \sigma A = \inf_n \frac{A(n)}{n}. \end{equation*} Schnirelmann density has the following properties:

1. $A(n)\geq n \sigma A$ for all $n$
2. $\sigma A=1$ if and only if $\mathbb{N}\subseteq A$
3. if $1$ does not belong to $A$ then $\sigma A=0$

Schnirelmann proved that if $0 \in A \cap B$ then \begin{equation*} \sigma(A+B)\geq \sigma A + \sigma B - \sigma A \cdot \sigma B \end{equation*}and also if $\sigma A + \sigma B \geq 1$ then $\sigma (A+B)=1$ From these he deduced that if $\sigma A>0$ then $A$ is an additive basis.