Combinatorial and Additive Number Theory IV

Let $\mathcal{B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets positive integers. $g_{\mathcal{B} }(n)$ count the number representations $n$ in form $n b_1\cdots b_h$, where $b_i \in B_i$ for all $i \{1,\ldots, h\}$. It is proved that $\liminf_{n\rightarrow \infty} g_{\mathcal{B} }(n) \geq 2$ implies $\limsup_{n\rightarrow \infty$.