Multiplicative complements, II.

Abstract

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 $\underset{x\to \infty }{\lim }\frac{A\left(x\right)B\left(x\right)}{x}=\infty$. We present some tight density bounds in connection with multiplicative complements.