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 $A(x)B(x)\mathrm{/}x\to 1$, then $A(x)B(x)-x\to \mathrm{\infty }$. 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.