On a Theorem of Plünnecke Concerning the Sum of A Basis and A Set of Positive Density

Abstract

In 1935 Erdős proved that every additive basis B of order h is an essential component by showing σ( A + B ) ≥ σ( A ) + (1/2 h) σ( A )(1 − σ( A )), where σ denotes Schnirelmann density. This lower bound was improved by Plünnecke to σ( A + B ) ≥ σ( A ) 1 − 1/ h , giving the best exponent. A simplified proof is presented.