 High-energy Roth theorem

For fixed $\theta >0$, a sufficiently large finite set of integers with additive energy at least $\theta |A{|}^3$ has a nonconstant three-term arithmetic progression. The Balog-Szemerédi-Gowers theorem and Ruzsa modelling lemma reduce the problem to the Roth theorem on three-term arithmetic progressions in a dense cyclic model.