Expanders and growth of normal subsets in finite simple groups of Lie type

We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $A$ of a finite simple group $G$ of Lie type of bounded rank, we either have $G\setminus \{1\}\subseteq A^2$ or $\mathrm{\mid }{A}^2\mathrm{\mid }\ge \mathrm{\mid }A{\mathrm{\mid }}^{1+\epsilon }$, for $\epsilon >0$. This improves a result of Gill, Pyber, Short and Szabó, and partially resolves a question of Pyber from the Kourovka notebook. We also propose a variant of Gowers’ trick for two subsets, and give applications to products of large subsets in groups of Lie type, improving some results of Larsen, Shalev and Tiep.