  We study the number of the vectors determined by two sets in d-dimensional
vector spaces over finite fields. We observe that the lower bound of
cardinality for the set of vectors can be given in view of an additive energy
or the decay of the Fourier transform on given sets. As an application of our
observation, we find sufficient conditions on sets where the Falconer distance
conjecture for finite fields holds in two dimension. Moreover, we give an
alternative proof of the theorem, due to Iosevich and Rudnev, that any Salem
set satisfies the Falconer distance conjecture for finite fields.
