The sum-product phenomenon in arbitrary rings

Details The sum-product phenomenon in arbitrary rings The sum-product phenomenon in arbitrary rings

2008-06-16

arxiv.org/abs/0806.2497v5

Mathematics

Combinatorics

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

Scientific paper

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.

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