A generalization of the diameter bound of Liebeck and Shalev for finite simple groups

Let $$G$$ be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there an absolute constant $$c$$ such whenever $$S$$ non-trivial normal subset in then $$S^{k} = G$$ for any integer $$k$$ at least $$c \cdot (\log|G|/\log|S|)$$ . This generalized by showing exists $$S_{1}$$ , $$\ldots $$ $$S_{k}$$ are subsets with $$\prod_{i=1}^{k} |S_{i}| \geq {|G|}^{c}$$ $$S_{1} \cdots S_{k}