On the volume set of point sets in vector spaces over finite fields

We show that if $\mathcal{E}$ is a subset of the $d$-dimensional vector space over a finite field $\text{mathbbm}{F}_q$ ($d\ge 3$) of cardinality $|$\mathcal{E}$| \geq (d-1)q^{d - 1}$,thenthesetofvolumesof$d$-dimensional parallelepipeds determined by $\mathcal{E}$ covers $\text{mathbbm}{F}_q$. This bound is sharp up to a factor of $(d-1)$ as taking $\mathcal{E}$ to be a $(d - 1)$-hyperplane through the origin shows.