  We extend previous work on Hrushovski's stabilizer's theorem and prove a
measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on
products of three types. This generalizes results of Gowers on products of
three sets and yields model-theoretic proofs of existing asymptotic results for
quasirandom groups. We also obtain a model-theoretic proof of Roth's theorem on
the existence of arithmetic progressions of length $3$ for subsets of positive
density in suitable definably amenable groups, such as countable amenable
abelian groups without involutions and ultraproducts of finite abelian groups
of odd order.
