Inversion and subspaces of a finite field

Abstract

Consider two F _q -subspaces A and B of a finite field, of the same size, and let A ⁻¹ denote the set of inverses of the nonzero elements of A. The author 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. Csajbók refined this to the following quantitative statement: if A ⁻¹ ⊈ B, then the bound |A ⁻¹∩B| ≤ 2|B|/q − 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q ³. Our main result is a proof of the stronger bound |A ⁻¹ ∩ B| ≤ |B|/q · (1 + O _d (q ^−1/2)), for |B| = q ^d with d > 3. We also classify all examples with |B| ≤ q ³ which attain equality or near-equality in Csajbók’s bound.