On function $SX$ of additive complements

Two sets $A$, $B$ of nonnegative integers are called <span class="italic">additive complements</span>, if all sufficiently large integers can be expressed as the sum of two elements from $A$ and $B$. We call $A$, $B$ <span class="italic">perfect additive complements</span> 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 Erdős 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.