On the distribution of the entries of a fixed-rank random matrix over a finite field

Let $r>0$ be an integer, let ${\mathbb{F}}_q$ be a finite field of $q$ elements, and let $\mathcal{A}$ be a nonempty proper subset of ${\mathbb{F}}_q$. Moreover, let $\mathbf{M}$ be a random $m\times n$ rank-$r$ matrix over ${\mathbb{F}}_q$ taken with uniform distribution. We prove, in a precise sense, that, as $m,n\to +\mathrm{\infty }$ and $r,q,\mathcal{A}$ are fixed, the number of entries of $\mathbf{M}$ that belong to $\mathcal{A}$ approaches a normal distribution.