On the elliptic curve analogue of the sum-product problem

Let E be an elliptic curve over a finite field F_q of q elements and x (P) to denote the x-coordinate of a point P = (x (P), y (P)) ∈ E. Let ⊕ denote the group operation in the Abelian group E (F_q) of F_q-rational points on E. We show that for any sets R, S ⊆ E (F_q) at least one of the sets{x (R) + x (S) : R ∈ R, S ∈ S} and {x (R ⊕ S) : R ∈ R, S ∈ S} is large. This question is motivated by a series of recent results on the sum-product problem over F_q.