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	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