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# 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 basis 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 order 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. Zbl 0498.10001.

Related:

SchnirlemannDensity, Basis2

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11B05*no label found*11B13

*no label found*

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