On the elliptic curve analogue of the sum-product problem

Let E be an elliptic curve over a finite field Fq 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(Fq) of Fq -rational points on E. We show that for any sets 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 Fq . © 2008 Elsevier Inc. All rights reserved.