Exact additive complements

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