On the elliptic curve analogue of the sum-product problem
The author studies the sum-product phenomenon in the context of elliptic curves over a finite field. Let \(E\) be an elliptic curve over a finite field \(\mathbb F_q\) of \(q\) elements and \(x(P)\) denotes the \(x\)-coordinate of a point \(P=(x(P),y(P))\in E\). Let \(\oplus\) denote the group operation in the abelian group \(E(\mathbb F_q)\) of \(\mathbb F_q\)-rational points on \(E\). This author proves that for any sets \(\mathcal R,\mathcal S\subseteq E(\mathbb F_q)\) at least one of the sets \[ \{x(R)+x(S): R\in \mathcal R,S\in\mathcal S\} \] and \[ \{x(R\oplus S): R\in \mathcal R,S\in\mathcal S\} \] is large. This paper is concerned with the sum-product problem, which is a classical one in additive number theory. The classical sum-product problem is the following: for any sets \(\mathcal A,\mathcal B\subseteq \mathbb F_q\), at least one of the sets \(\{a+b: a\in \mathcal A,b\in B\}\) and \(\{ab: a\in \mathcal A,b\in \mathcal B\}\) is large.
scientific article; zbMATH DE number 5316924
scientific article