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 $A^{-1}\subseteq ̸B$, then the bound $\mathrm{\mid }{A}^{-1}\cap B\mathrm{\mid }\le 2\mathrm{\mid }B\mathrm{\mid }\mathrm{/}q-2$ holds. He also gave examples showing that his bound is sharp for $\mathrm{\mid }B\mathrm{\mid }\le q^3$. 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 $\mathrm{\mid }B\mathrm{\mid }\le q^3$ which attain equality in Csajb'{o}k’s bound.