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

1.
$A(n)\ge n\sigma A$ 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
$$\sigma (A+B)\ge \sigma A+\sigma B\sigma A\cdot \sigma B$$ 
and also if $\sigma A+\sigma B\ge 1$, then $\sigma (A+B)=1$. From these he deduced that if $\sigma A>0$ then $A$ is an additive basis.
Title  Schnirelmann density^{} 
Canonical name  SchnirelmannDensity 
Date of creation  20130322 13:19:36 
Last modified on  20130322 13:19:36 
Owner  bbukh (348) 
Last modified by  bbukh (348) 
Numerical id  9 
Author  bbukh (348) 
Entry type  Definition 
Classification  msc 11B13 
Classification  msc 11B05 
Synonym  Shnirel’man density 
Synonym  Shnirelman density 
Related topic  Basis2 
Related topic  EssentialComponent 
Related topic  MannsTheorem 