On a question of Cochrane and Pinner concerning multiplicative subgroups

Abstract

Answering a question of T. Cochrane and C. Pinner, we prove that for any {\epsilon}>0, sufficiently large prime number p and an arbitrary multiplicative subgroup R of the field Z/pZ, p^{\epsilon} < |R| < p^{2/3-{\epsilon}} the following holds |R + R|, |R - R| > |R|^{3/2+{\delta}}, where {\delta}>0 depends on {\epsilon} only.