We extend previous work on Hrushovski’s stabilizer 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
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.
Keywords
model theory, additive combinatorics, arithmetic
progressions, quasirandom groups