2 problems

Let $r$ be a positive integer, let $G$ be a finite simple group of Lie type of rank $r$, and let $S$ be a generating set of $G$. For $g\in G$, write $S^g=g^{-1}Sg$, and for a posit…

Let $G$ be a finite simple group, let $S$ be a subset of $G$, and write $S^g=g^{-1}Sg$. Product-growth conjecture. There exist $varepsilon>0$ and a positive integer $b$ such that,…