Exact additive complements

Abstract

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