Cayley Digraphs of Matrix Rings over Finite Fields

TL;DR

This paper explores the spectral properties of Cayley digraphs on matrix rings over finite fields, demonstrating sum representations of matrices, sum-product phenomena, and classical unit sum results in finite rings.

Contribution

It introduces new spectral analyses of unit-graphs on matrix rings, establishes sum representations of matrices, and proves sum-product results over finite fields.

Findings

Every nonzero matrix over _q is a sum of two SL_n matrices for n>1.

Eigenvalues of the graphs are expressed via Kloosterman sums.

Large subsets of matrix rings contain matrices with differences of any given determinant.

Abstract

We use the \emph{unit-graphs} and the \emph{special unit-digraphs} on matrix rings to show that every $n \times n$ nonzero matrix over $\Bbb F_q$ can be written as a sum of two $\operatorname{SL}_n$-matrices when $n>1$. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if $X$ is a subset of $\operatorname{Mat}_2 (\Bbb F_q)$ with size $|X| > \frac{2 q^3 \sqrt{q}}{q - 1}$, then $X$ contains at least two distinct matrices whose difference has determinant $\alpha$ for any $\alpha \in \Bbb F_q^{\ast}$. Using this result we also prove a sum-product type result: if $A,B,C,D \subseteq \Bbb F_q$ satisfy $\sqrt[4]{|A||B||C||D|}= \Omega (q^{0.75})$ as $q \rightarrow \infty$, then $(A - B)(C - D)$ equals all of $\Bbb F_q$. In particular, if $A$ is a subset of $\Bbb F_q$ with cardinality $|A| > \frac{3} {2} q^{\frac{3}{4}}$, then the subset $(A - A) (A - A)$ equals all of $\Bbb F_q$. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.