Analysis reveals large sumsets or product sets in finite sets within arbitrary rings, highlighting zero-divisor impacts.
The 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⋅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 and cyclic groups /q. 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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Terence Tao (2009) studied this question.
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