On a property of additive complements

Two infinite sequences $A$ and $B$ are called infinite additive complements if every sufficiently large integer can be expressed as the sum of two elements taken from $A$ and $B$. Let $A(x)$ (resp. $B(x)$) be the number of elements in $A$ (resp. $B$) not exceeding $x$. Motivated by a recent result [3], the authors proved that, for infinite additive complements $A$, $B$, if $\limsup\frac{A(2x)B(2x)}{A(x)B(x)}\!＜\!2$ or $\limsup\frac{A(2x)B(2x)}{A(x)B(x)}\!＞\!4$, then $A(x)B(x)-x\rightarrow+\infty$ as $x\to+\infty$. Furthermore, the above constants $2$ and $4$ cannot be improved.