Inversion and subspaces of a finite field

Let $A$ and $B$ two $F_q$-subspaces of a finite field, of the same size, and let $A^{-1}$ denote the set of inverses of the nonzero elements of $A$. Mattarei proved that $A^{-1}$ can only be contained in $A$ if either $A$ is a subfield, or $A$ is the set of trace zero elements in a quadratic extension of a field. Csajb\'{o}k refined this to the following quantitative statement: if $A^{-1}\not\subseteq B$, then the bound $|A^{-1}\cap B|\le 2|B|/q-2$ holds. He also gave examples showing that his bound is sharp for $|B|\le q^3$. Our main result is a proof of the stronger bound $|A^{-1}\cap B|\le |B|/q\cdot\bigl(1+O_d(q^{-1/2})\bigr)$, for $|B|=q^d$ with $d>3$. We also classify all examples with $|B|\le q^3$ which attain equality in Csajb\'{o}k's bound.