Exact Additive Complements

Abstract

Let $A,B$ be sets of positive integers such that $A+B$ contains all but finitely many positive integers. Sarkozy and Szemeredi proved that if $ A(x)B(x)/x \to 1$, then $A(x)B(x)-x \to \infty $. Chen and Fang considerably improved Sarkozy and Szemeredi's bound. We further improve their estimate and show by an example that our result is nearly best possible.