Distribution of determinant of sum of matrices

Let $mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $det S$ for certain types of subsets $S$ in the ring $M_2(mathbb F_q)$ of $2times 2$ matrices with entries in $mathbb F_q$. For $iin mathbb{F}_q$, let $D_i$ be the subset of $M_2(mathbb F_q)$ defined by $ D_i := {xin M_2(mathbb F_q): det(x)=i}.$ Then our results can be stated as follows. First of all, we show that when $E$ and $F$ are subsets of $D_i$ and $D_j$ for some $i, jin mathbb{F}_q^*$, respectively, we have $$det(E+F)=mathbb F_q,$$ whenever $|E||F|ge {15}^2q^4$, and then provide a concrete construction to show that our result is sharp. Next, as an application of the first result, we investigate a distribution of the determinants generated by the sum set $(Ecap D_i) + (Fcap D_j),$ when $E, F$ are subsets of the product type, i.e., $U_1times U_2subseteq mathbb F_q^2times mathbb F_q^2$ under the identification $ M_2(mathbb F_q)=mathbb F_q^2times mathbb F_q^2$. Lastly, as an extended version of the first result, we prove that if $E$ is a set in $D_i$ for $i e 0$ and $k$ is large enough, then we have [det(2kE):=det(underbrace{E + dots + E}_{2k~terms})supseteq mathbb{F}_q^*,] whenever the size of $E$ is close to $q^{frac{3}{2}}$. Moreover, we show that, in general, the threshold $q^{frac{3}{2}}$ is best possible. Our main method is based on the discrete Fourier analysis.