Locally compact models for approximate rings

By an 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 additive translates of $X$. We prove that each approximate subring $X$ of a ring has a locally compact model, i.e. a ring homomorphism $f \colon \langle X \rangle \to S$ for some locally compact ring $S$ such that $f[X]$ is relatively compact in $S$ and there is a neighborhood $U$ of $0$ in $S$ with $f^{-1}[U] \subseteq 4X + X \cdot 4X$ (where $4X:=X+X+X+X$). This $S$ is obtained as the quotient of the ring $\langle X \rangle$ interpreted in a sufficiently saturated model by its type-definable ring connected component. The above theorem can be seen as a general structural result about approximate subrings: every approximate subring $X$ can be recovered up to additive commensurability as the preimage by a locally compact model $f \colon \langle X \rangle \to S$ of any relatively compact neighborhood of $0$ in $S$. It also leads to more precise structural or even classification results. For example, we deduce that every [definable] approximate subring $X$ of a ring of positive characteristic is additively commensurable with a [definable] subring contained in $4X + X \cdot 4X$. This implies that for any given $K,L \in \mathbb{N}$ there exists $C(K,L)$ such that every $K$-approximate subring $X$ (i.e. $K$ additive translates of $X$ cover $X \cdot X \cup (X+X)$) of a ring of positive characteristic $\leq L$ is additively $C(K,L)$-commensurable with a subring contained in $4X + X \cdot 4X$. We also deduce a classification of finite approximate subrings of rings without zero divisors: for every $K \in \mathbb{N}$ there exists $N(K) \in \mathbb{N}$ such that for every finite $K$-approximate subring $X$ of a ring without zero divisors either $|X| <N(K)$ or $4X + X \cdot 4X$ is a subring which is additively $K^{11}$-commensurable with $X$.