On growth of the number of determinants with restricted entries

Abstract

Let $A$ be a finite subset of a field $\mathbb{F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$D_n(A)=\{D\in \mathbb{F}|\exists a_{ij}\in A,1\le i,j\le n,\det ((a_{ij}))=D\},$$ where the symbol $(a_{ij})$ defines the matrix with elements $a_{ij}$. How big is the size of the set $D_n(A)$ comparing to the size of the set $A$?