The sum-product phenomenon in arbitrary rings

TL;DR

This paper extends the sum-product phenomenon to arbitrary rings, including non-commutative and rings without identity, providing new formulations and generalizations of existing sum-product theorems.

Contribution

It introduces rigorous formulations of the sum-product phenomenon in arbitrary rings with few zero-divisors, broadening the scope beyond well-studied rings like reals and cyclic groups.

Findings

Recovered existing sum-product theorems in new contexts

Generalized sum-product results to non-commutative rings

Provided formulations applicable to rings without identity

Abstract

The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $\R$ and cyclic groups $\Z/q\Z$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sum-product phenomenon in such rings in the case when $A$ encounters few zero-divisors of $R$. As applications we recover (and generalise) several sum-product theorems already in the literature.