Expanders on matrices over a finite chain ring, I

TL;DR

This paper investigates the expansion properties of matrices over finite chain rings, establishing sum-product estimates and demonstrating that certain matrix operations act as moderate expanders, extending recent results in the field.

Contribution

It introduces new sum-product estimates for matrices over finite chain rings and shows that the operation x+yz is a moderate expander with a specific exponent, generalizing prior work.

Findings

Sum-product estimate for matrices over finite chain rings

x+yz acts as a moderate expander with exponent (n+1)/6

Results extend previous theorems by Xie and Ge

Abstract

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is showed that $x+yz$ is a moderate expander on $n\times n$ matrices with exponent $\frac{n+1}{6}$. These results generalise the main theorems in a recent work of Xie and Ge. The proofs use spectral graph theory and elementary divisor theory.