essential component

If $A$ is a set of nonnegative integers such that

$$\sigma (A+B)>\sigma B$$(1)

for every set $B$ with Schnirelmann density , then $A$ is an essential component.

Erdős proved that every http://planetmath.org/node/3831basis is an essential component. In fact he proved that

$$\sigma (A+B)\ge \sigma B+\frac{1}{2h}(1-\sigma B)\sigma B,$$

where $h$ denotes the http://planetmath.org/node/3831order of $A$.

Plünnecke improved that to

$$\sigma (A+B)\ge \sigma B^{1-1/h}.$$

There are non-basic essential components. Linnik constructed non-basic essential component for which $A(n)=O(n^{ϵ})$ for every $ϵ>0$.