On some multiplicative properties of large difference sets

In our paper we study multiplicative properties of difference sets $A-A$ for large sets $A\subseteq \mathbb{Z}\mathrm{/}q\mathbb{Z}$ in the case of composite $q$. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.