Distribution of determinant of matrices with restricted entries over finite fields

For a prime power $q$, we study the distribution of determinent of matrices with restricted entries over a finite field $\text{mathbbm}{F}_q$ of $q$ elements. More precisely, let $N_d(\mathcal{A};t)$ be the number of $d\times d$ matrices with entries in $\mathcal{A}$ having determinant $t$. We show that $$N_d(\mathcal{A};t)=(1+o(1))\frac{\mathrm{\mid }\mathcal{A}{\mathrm{\mid }}^{d^2}}{q},$$ if $\mathrm{\mid }\mathcal{A}\mathrm{\mid }=\omega (q^{\frac{d}{2d-1}})$, $d⩾4$. When $q$ is a prime and $\mathcal{A}$ is a symmetric interval $[-H,H]$, we get the same result for $d⩾3$. This improves a result of Ahmadi and Shparlinski (2007).