Cayley Digraphs Associated to Arithmetic Groups

TL;DR

This paper connects number theory, combinatorics, and geometry through Cayley digraphs, demonstrating that matrices over finite fields can be expressed as sums of orthogonal matrices with bounds depending on field properties.

Contribution

It introduces a novel framework linking diverse mathematical areas and proves that matrices over finite fields are sums of a bounded number of orthogonal matrices, depending only on dimension and field congruence.

Findings

Every matrix in Mat_d(F_q) is a sum of a finite number of orthogonal matrices.

The number of orthogonal matrices needed depends only on d and the congruence class of q mod 4.

The results unify concepts across number theory, additive combinatorics, and geometric combinatorics.

Abstract

We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-S\'{a}rk\"{o}zy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\mathbb{F}_q), d \geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of the matrix, and on whether $q$ is congruent to $1$ or $3$ (mod $4$), but independent of $q$ otherwise.