# 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 $A(n)=O({n}^{\u03f5})$ for every $\u03f5>0$.

## 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.

Title | essential component |
---|---|

Canonical name | EssentialComponent |

Date of creation | 2013-03-22 13:19:42 |

Last modified on | 2013-03-22 13:19:42 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 7 |

Author | bbukh (348) |

Entry type | Definition |

Classification | msc 11B05 |

Classification | msc 11B13 |

Related topic | SchnirlemannDensity |

Related topic | Basis2 |