On the Gowers trick for classical simple groups

Abstract

If A, B, C are subsets in a finite simple group of Lie type G at least two of which are normal with $\left|A\right|\left|B\right|\left|C\right|$ relatively large, then we establish a stronger conclusion than $ABC=G$. This is related to a theorem of Gowers and is a generalization of a theorem of Larsen, Shalev, Tiep and the second author and Pyber.