On the structure of approximate rings

TL;DR

This paper develops a structure theorem for finite approximate subrings, revealing nilpotent quotients as key obstructions to growth, and extends sum-product phenomena to arbitrary rings.

Contribution

It introduces a general framework for sum-product phenomena in rings, identifying nilpotent quotients as fundamental obstructions and generalizing classical results.

Findings

Nilpotent quotients are the main obstructions to growth in approximate subrings.

Established a ring-theoretic analogue of Gromov's polynomial growth theorem.

Extended the structure theorem to approximate subrings of semi-simple real algebras.

Abstract

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate subrings. Our aim is to develop a general framework for the sum-product phenomenon that applies uniformly across arbitrary rings. The main result identifies nilpotent quotients as the fundamental obstruction to growth under both addition and multiplication. Another application of the main structure theorem is a ring-theoretic counterpart of Gromov's theorem on groups of polynomial growth. The principal tool in the proof is the existence of definable locally compact models for arbitrary approximate subrings from [Kru24]. This existence theorem extends beyond the finite (and pseudofinite) setting. To illustrate the scope of the method, we also establish a structure theorem for uniformly discrete approximate subrings of semi-simple real algebras, generalizing a classical sum-product result of Meyer.