In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?

PropositionThere exists an absolute constant $c$ such that, if $A$ is any symmetric subset of a finite field $\mathbb{F}_q$ with $|A|\geq 2$, then there are elements $g_1,\dots, g_n\in \mathbb{F}_q$ with $$Ag_1+\cdots Ag_n = \mathbb{F}_q$$ and $n\leq c\log(q)/\log|A|$.

Please note that I haven't checked every detail of the proof! I think it works but you never know... I also haven't tried to optimise the constant $c$ but it seems like $34$ works. Finally, although I need the fact that $A$ is symmetric for my proof, I would imagine that the same bound holds without this (symmetric here means that $A=-A$).

When I say "is this interesting?" I mean, first, that I'd like to know whether this proposition is easy-peasy for any one with a modicum of knowledge in algebraic combinatorics and, if this is not the case, whether there might be any applications.

**Background**: I stumbled on this result while trying to prove a statement about width in finite simple groups. There is a famous conjecture in this area due to Liebeck, Nikolov and Shalev ("The Product Decomposition Conjecture") which says the following:

Conjecture: There exists an absolute constant $c$ such that if $G$ is a finite simple group and $S$ is a subset of $G$ of size at least two, then $G$ is a product of $n$ conjugates of $S$, and $n \leq c \log|G|/ \log |S|$.

The proposition I state at the top is basically the same statement but for the group $G=\mathbb{F}_q\rtimes\mathbb{F}_q^*$ (where we restrict $A$ to be in the normal subgroup $\mathbb{F}_q$).

One of the things that surprises me is that one (conjecturally) has a bound on the width of two families of groups that look completely different - i.e. finite simple groups, and $\mathbb{F}_q\rtimes\mathbb{F}_q^*$. Of course these families are also the settings in which the most famous growth statements have been proven (cf. work of Helfgott, Pyber, Szabo, Breuillard, Green, Tao, Bourgain, Katz and many others), so perhaps this is not surprising. Note, though, that the bounds connected to width (i.e. multiplication by conjugates) are much stronger than those given by growth so the connection between the two is not entirely obvious...