Let $V$ be a finite dimensional vector space over a finite field $\mathbb{F}_q$ (the most interesting case to me is $q=O(1)$). Let $A$ be an arbitrary subset of $V$ that spans $V$ over $\mathbb{F}_q$. Let $\mu=|A|/|V|$ be the density of $A$. What can we say about the smallest $k$ such that every $v\in V$ can be written as a linear combination of $t\leq k$ elements $a_1,\dots, a_t\in A$ $$ v=\sum_{i=1}^t c_i a_i $$ with coefficients $c_i\in\mathbb{F}_q$?

I can prove $k\leq (1/\mu)^c$ for some constant $c>0$, basically by repeating applying the fact that $|A-A|\geq (3/2)|A|$ if $A$ is not a subgroup of $V$. This fact follows from an argument of Laba, and is stated as Lemma 2.5 in the following survey by Lovett: http://cseweb.ucsd.edu/~slovett/files/addcomb-survey.pdf

Is there a better upper bound? And any lower bounds known?