Arithmetic Combinatorics

We will study the following problem: Given $n$ subsets $A_1, A_2,\ldots, A_n\subset \mathbb{F}_q$ of a finite field $\mathbb{F}_q$ with $q$ elements. Let $|A_1|\cdot |A_2|\cdot\ldots dot|A_n|>q^{1+\varepsilon}$ for some $\varepsilon>0,$ one needs to find a natural number $N(n,\varepsilon)$ such that $NA_1A_2\ldots A_n=\mathbb{F}_q.$ Several results of this type will be presented.