On additive complements

Two infinite sequences $A$ and $B$ of non-negative integers are called additive complements if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of $A$ and $B$. For additive complements $A$ and $B$, Sárközy and Szemerédi proved that if $\limsup\limits_{x\rightarrow\infty}\frac{A(x)B(x)}{x}\le 1$, then $A(x)B(x)-x\rightarrow+\infty$. In this paper, we prove that for additive complements $A$ and $B$, if $\limsup\limits_{x\rightarrow\infty}\frac{A(x)B(x)}{x}<\frac 54$ or $\limsup\limits_{x\rightarrow\infty}\frac{A(x)B(x)}{x}>2$, then $A(x)B(x)-x\rightarrow+\infty$.