Linear subspaces of finite fields with large inverse-closed subsets

We call a subset of a finite field inverse-closed, if it is closed with respect to taking inverses. Our goal is to prove that an additive subgroup of a finite field with a large inverse-closed subset is necessarily inverse-closed. Actually, this is obtained as the special case A=B and q=p of the following more general result: Let A and B be linear subspaces of a finite field of characteristic p, considered as vector spaces over the subfield of order q, with the same dimension. If the set of inverses of the non-zero elements of A shares at least 2|B|/q-1 elements with B, then they are both one-dimensional subspaces over the same subfield. In the special case q=2, the above result holds under a weaker condition. We exhibit some examples showing sharpness when |A|≤q^3 and give some characterizations and geometric descriptions of these examples. Similar results are stated for infinite fields.