Multiplicative complements II

In this paper we prove that if $A$ and $B$ are infinite subsets of positive integers such that every positive integer $n$ can be written as $n=ab$, $a\in A$, $b\in B$, then ${\displaystyle \underset{x\to \mathrm{\infty }}{\lim }\frac{A(x)B(x)}{x}=\mathrm{\infty }}$. We also prove many other results about sets like this.