Submission date: 4 January 2007

Abstract:

Generalising results of Chatzidakis, van den Dries and Macintyre for finite fields, a notion of `asymptotic class' of finite structures is introduced. An ultraproduct of an asymptotic class is supersimple and will have a definable `measure' on the definable sets, which leads to the notion of `measurable structure'. The paper investigates asymptotic classes which are one-dimensional (there are generalisations to higher dimension due to Elwes). Apart from finite fields, examples include Paley graphs, cyclic groups, envelopes of smoothly approximable Lie geometries. Among results on measurable structures, it is shown that a one-dimensional measurable group is finite-by- abelian-by-finite.

Mathematics Subject Classification: 03C13, 03C45

Keywords and phrases: asymptotic class, measurable structure, smoothly approximable structure, pseudofinite field, supersimple theory

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