The sum-product phenomenon in arbitrary rings

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 $\mathbb{R}$ and cyclic groups $\mathbb{Z}/q\mathbb{Z}$. 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.

26 pages, no figures, to appear, Contributions to Discrete Mathematics. Some final corrections

References in corpus (5)

• An explicit sum-product estimate in ${\mathbb{F}}_p$
• An upper bound on the multiplicative energy
• The sum-product estimate for large subsets of prime fields
• Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture
• Pinned distance sets, Wolff's exponent in finite fields and improved sum-product estimates

Cited by in corpus (2)

• Growth in SL_3(Z/pZ)
• Sums and products along sparse graphs