On the elliptic curve analogue of the sum-product problem

Let E be an elliptic curve over a finite field FqFq of q   elements and x(P)x(P) to denote the x  -coordinate of a point P=(x(P),y(P))∈EP=(x(P),y(P))∈E. Let ⊕ denote the group operation in the Abelian group E(Fq)E(Fq) of FqFq-rational points on E. We show that for any sets R,S⊆E(Fq)R,S⊆E(Fq) 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 FqFq.