  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. In this paper, we extend the
results of Liu and Fang in 2016 and obtain some results on additive
complements. For example, we prove that there exist additive complements $A$
and $B$ such that $\limsup_{x\to+\infty} A(x)B(x)/x= 2$ and $A(x)B(x) - x = 1$
for infinitely positive integers $x$.
