High-energy Roth theorem
= 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.