Writing finite simple groups of Lie type as products of subset conjugates

The Liebeck-Nikolov-Shalev conjecture [LNS12] asserts that, for any finite simple non-abelian group $G$ and any set $A\subseteq G$ with $\mathrm{\mid }A\mathrm{\mid }\ge 2$, $G$ is the product of at most $N\frac{\log \mathrm{\mid }G\mathrm{\mid }}{\log \mathrm{\mid }A\mathrm{\mid }}$ conjugates of $A$, for some absolute constant $N$. For $G$ of Lie type, we prove that for any $\epsilon >0$ there is some $N_{\epsilon }$ for which $G$ is the product of at most $N_{\epsilon }{\left(\frac{\log \mathrm{\mid }G\mathrm{\mid }}{\log \mathrm{\mid }A\mathrm{\mid }}\right)}^{1+\epsilon }$ conjugates of either $A$ or $A^{-1}$. For symmetric sets, this improves on results of Liebeck, Nikolov, and Shalev [LNS12] and Gill, Pyber, Short, and Szabó [GPSS13]. During the preparation of this paper, the proof of the Liebeck-Nikolov-Shalev conjecture was completed by Lifshitz [Lif24]. Both papers use [GLPS24] as a starting point. Lifshitz’s argument uses heavy machinery from representation theory to complete the conjecture, whereas this paper achieves a more modest result by rather elementary combinatorial arguments.