(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?