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 |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.