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