# 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

 $\sigma A=\inf_{n}\frac{A(n)}{n}.$

has the following properties:

1. 1.

$A(n)\geq n\sigma A$ for all $n$.

2. 2.

$\sigma A=1$ if and only if $\mathbb{N}\subseteq A$

3. 3.

if $1$ does not belong to $A$, then $\sigma A=0$.

Schnirelmann proved that if $0\in A\cap B$ then

 $\sigma(A+B)\geq\sigma A+\sigma B-\sigma A\cdot\sigma B$

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.

 Title Schnirelmann density Canonical name SchnirelmannDensity Date of creation 2013-03-22 13:19:36 Last modified on 2013-03-22 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