On the Gowers trick for classical simple groups

If $A$, $B$, $C$ are subsets in a finite simple group of Lie type $G$ at least two of which are normal with $\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }\mathrm{\mid }C\mathrm{\mid }$ 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.