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Generalizing results for approximate subgroups, we study approximate equivalence
relations up to commensurability, in the presence of a definable measure.
As a basic framework, we give a presentation of probability logic based on
continuous logic. Hoover’s normal form is valid here; if one begins with a discrete
logic structure, it reduces arbitrary formulas of probability logic to correlations
between quantifier-free formulas. We completely classify binary correlations in terms
of the Kim–Pillay space, leading to strong results on the interpretative power of pure
probability logic over a binary language. Assuming higher amalgamation
of independent types, we prove a higher stationarity statement for such
correlations.
We also give a short model-theoretic proof of a categoricity theorem for
continuous logic structures with a measure of full support, generalizing theorems of
Gromov–Vershik and Keisler, and often providing a canonical model for a complete
pure probability logic theory. These results also apply to local probability logic,
providing in particular a canonical model for a local pure probability logic theory
with a unique 1-type and geodesic metric.
For sequences of approximate equivalence relations with an “approximately
unique” probability logic 1-type, we obtain a structure theorem generalizing
the “Lie model” theorem for approximate subgroups (Theorem 5.5). The
models here are Riemannian homogeneous spaces, fibered over a locally finite
graph.
Specializing to definable graphs over finite fields, we show that after a finite
partition, a definable binary relation converges in finitely many self-compositions to
an equivalence relation of geometric origin. This generalizes the main lemma for
strong approximation of groups.
For NIP theories, pursuing a question of Pillay’s, we prove an archimedean
finite-dimensionality statement for the automorphism groups of definable
measures, acting on a given type of definable sets. This can be seen as an
archimedean analogue of results of Macpherson and Tent on NIP profinite
groups.
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