Distribution of the determinants of sums of matrices

Abstract

Let $\mathbb{F}\_q$ be an arbitrary finite field of order $q$ and let $M\_2(\mathbb F\_q)$ be the ring of all $2\times 2$ matrices with entries in $\mathbb F\_q$. In this article, we study $\mathrm{det} S$ for certain types of subsets $S \subset M\_2(\mathbb F\_q)$. For $i\in \mathbb{F}\_q$, let $D\_i$ be the subset of $M\_2(\mathbb F\_q)$ defined by $D\_i := {x\in M\_2(\mathbb F\_q): \det(x)=i}.$ We first show that when $E$ and $F$ are subsets of $D\_i$ and $D\_j$ for some $i, j\in \mathbb{F}\_q^\*$, respectively, we have $$ \mathrm{det}(E+F)=\mathbb F\_q $$ whenever $|E| |F|\ge {15}^2q^4$, and then provide a concrete construction to show that our result is sharp. Secondly, as an application of the first result, we investigate the 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 $$ \mathrm{det}(2kE):=\det(\underbrace{E + \dots + E}\_{2k : \mathrm{terms}}) \supseteq \mathbb{F}\_q^\* $$ whenever the size of $E$ is close to $q^{3/2}$. Moreover, we show that, in general, the threshold $q^{3/2}$ is the best possible. Our methods are based on discrete Fourier analysis.