Complete type amalgamation for nonstandard finite groups

We extend previous work on Hrushovski's stabilizer theorem and prove a measuretheoretic 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.