On the structure of approximate rings

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.