  Two sets $A,B$ of nonnegative integers are called \emph{additive
complements}, if all sufficiently large integers can be expressed as the sum of
two elements from $A$ and $B$. We further call $A,B$ \emph{perfect additive
complements} if every nonnegative integer can be uniquely expressed as the sum
of two elements from $A$ and $B$. Let $A(x)$ be the counting function of $A$.
In this paper, we focus on the function $SX$, where
$SX=\limsup_{x\rightarrow\infty}\frac{\max\{A(x),B(x)\}}{\sqrt{x}}$ was
introduced by Erd\H{o}s and Freud in 1984. As a main result, we determine the
value of $SX$ for perfect additive complements and further fix the infimum. We
also give the absolute lower bound of $SX$ for additive complements.
