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 $0<\sigma B<1$ , 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)\geq\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)\geq\sigma B^{1-1/h}.

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

References

1 Heini Halberstam and Klaus Friedrich Roth. Sequences. Springer-Verlag, second edition, 1983. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0498.10001Zbl 0498.10001.