# Expansion of product of simple Lie group

(a quite technical question if you want to skip).

I am looking at the paper Breuillard, Green, Guralnick, and Tao - Expansion in finite simple groups of Lie type; Specifically, proposition 8.4.

Proposition $$8.4 .$$ Let $$r \in \mathbf{N}$$ and $$\varepsilon>0 .$$ Suppose $$G=G_{1} G_{2},$$ where $$G_{1}$$ and $$G_{2}$$ are products of at most $$r$$ finite simple (or quasisimple) groups of Lie type of rank at most r. Suppose that no simple factor of $$G_{1}$$ is isomorphic to a simple factor of $$G_{2} .$$ If $$x_{1}=$$ $$x_{1}^{(1)} x_{1}^{(2)}, \ldots, x_{k}=x_{k}^{(1)} x_{k}^{(2)}$$ are chosen so that $$\left\{x_{1}^{(1)}, \ldots, x_{k}^{(1)}\right\}$$ and $$\left\{x_{1}^{(2)}, \ldots, x_{k}^{(2)}\right\}$$ are both $$\varepsilon$$-expanding generating subsets in $$G_{1}$$ and $$G_{2}$$ respectively, then $$\left\{x_{1}, \ldots, x_{k}\right\}$$ is $$\delta$$. expanding in G for some $$\delta=\delta(\varepsilon, r)>0$$

There are two cases. The first one is OK. In the second one, $$\lvert G_2\rvert \ge \lvert G_1\rvert^{\beta/5}$$. In this case, $$m = O_c(log \lvert G\rvert)$$. And $$\delta$$ seems constant, indepedent of $$\epsilon$$.

My question is whether there is anything we can say about $$\delta$$ besides $$\delta>0$$. For example, $$\delta>1/2$$ would be nice, or $$\delta> \lvert G_1\rvert/\lvert G_2\rvert$$ or $$\delta > C\epsilon$$ (that would be best).

To put it into more easy to handle terms, I don't get how $$m$$ is obtained given a $$\lvert G\rvert^\nu$$-approximate group and using the weighted Balog-Szemerédi-Gowers.

Another question which I don't understand is: it is implied that $$(1/2-\beta/5) > 0$$ in the first case. Why is that?

• All four of the authors of the paper are alive and hopefully well. Why don't you ask them directly? – user6976 Mar 18 at 0:58