The central limit theorem for entries of random matrices with specific rank over finite fields

Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathcal{A}$ a non-empty proper subset of $\mathbb{F}_q$. Let $\mathbf{M}$ be a random $m \times n$ matrix of rank $r$ over $\mathbb{F}_q$ taken with uniform distribution. It was proved recently by Sanna that as $m,n \to \infty$ and $r,q,\mathcal{A}$ are fixed, the number of entries of $\mathbf{M}$ in $\mathcal{A}$ approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when $r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we answer this question affirmatively.