A theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if $\phi(x,y)$ is a formula in the language of rings (where $x,y$ are tuples) then the size of the solution set of $\phi(x,a)$ in any finite field $F_q$ (where $a$ is a parameter tuple from $F_q$) takes one of finitely many dimension-measure pairs as $F_q$ and a vary: roughly, for a finite set $E$ of pairs $(\mu,d)$ ($\mu$ rational, $d$ integer) dependent on $\phi$, any set $\phi(F_q,a)$ has size roughly $\mu.q^d$ for some $(\mu,d) \in E$. It follows, for example, that there is no single formula $\phi(x,y)$ such that in every finite field $F_{p^2}$ there is a tuple $a$ in $F_{p^2}$ such that $\phi(F_{p^2},a)$ is the prime subfield $F_p$.
This led in work of Elwes, Steinhorn and myself to the notion of `asymptotic class� of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). There is a corresponding notion for infinite structures of `measurable structure� (e.g. a pseudofinite field, by the Chatzidakis- van den Dries-Macintyre theorem)). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. I will discuss a body of work in progress with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a far richer range of examples with fewer model-theoretic constraints (for example, the corresponding infinite `generalised measurable� structures need no longer have simple theory, but is NSOP, that is, it does not have the strict order property.
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