essential component

If A is a set of nonnegative integers such that

σ(A+B)>σB (1)

for every set B with Schnirelmann densityMathworldPlanetmath 0<σB<1, then A is an essential component.

Erdős proved that every is an essential component. In fact he proved that


where h denotes the of A.

Plünnecke improved that to


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


  • 1 Heini Halberstam and Klaus Friedrich Roth. Sequences. Springer-Verlag, second edition, 1983. 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