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If $A$ is a set of nonnegative integers such that \begin{equation} \sigma(A+B)>\sigma B \end{equation}for every set $B$ with Schnirelmann density $0<\sigma B<1$ , then $A$ is an essential component.
Erdos proved that every basis is an essential component. In fact he proved that \begin{equation*} \sigma(A+B)\geq \sigma B+\frac{1}{2h}(1-\sigma B)\sigma B, \end{equation*}where $h$ denotes the order of $A$ .
Plünnecke improved that to \begin{equation*} \sigma(A+B)\geq \sigma B^{1-1/h}. \end{equation*} There are non-basic essential components. Linnik constructed non-basic essential component for which $A(n)=O(n^\epsilon)$ for every $\epsilon>0$ .
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- Heini Halberstam and Klaus Friedrich Roth.
Sequences.
Springer-Verlag, second edition, 1983.
Zbl 0498.10001.
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