Distribution of determinant of sum of matrices

Abstract

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 $2\times 2$ matrices with entries in ${\mathbb{F}}_q$. For $i\in {\mathbb{F}}_q$, let $D_i$ be the subset of $M_2({\mathbb{F}}_q)$ defined by $D_i:=\{x\in M_2({\mathbb{F}}_q):\det (x)=i\}.$ 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 $i,j\in {\mathbb{F}}_q^{\ast }$, respectively, we have $$\det (E+F)={\mathbb{F}}_q,$$ whenever $|E||F|\ge {15}^2{q}^4$, 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 $(E\cap D_i)+(F\cap D_j),$ when $E,F$ are subsets of the product type, i.e., $U_1\times U_2\subseteq {\mathbb{F}}_q^2\times {\mathbb{F}}_q^2$ under the identification $M_2({\mathbb{F}}_q)={\mathbb{F}}_q^2\times {\mathbb{F}}_q^2$. Lastly, as an extended version of the first result, we prove that if $E$ is a set in $D_i$ for $i\ne 0$ 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.