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August 17, 2025

The sum-product phenomenon in arbitrary rings

Puntos clave

The research shows that in a finite set within an arbitrary ring, large sumsets or product sets are expected unless a specific structure is present.
Key evidence indicates that the presence of few zero-divisors allows for rigorous formulations of the sum-product phenomenon in generalized settings.
Utilizing abstract mathematical techniques, the study broadens the understanding of sum-product behavior beyond classical structures, incorporating various ring types.
The work implies potential applications in diverse mathematical fields, necessitating further exploration of the sum-product phenomenon in different algebraic contexts.

Resumen

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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