Additive properties of multiplicative subgroups of \(\mathbb F_{p}\) (Q2907044)

By a result of \textit{D. R. Heath-Brown} and \textit{S. Konyagin} [Q. J. Math. 51, No. 2, 221--235 (2000; Zbl 0983.11052)] it is known that for any multiplicative subgroup \(R \subset \mathbb{F}_p^{\ast}\) with \(|R| = O(p^{2/3})\) one has \(|R\pm R| \gg |R|^{3/2}\); as usual \(\mathbb{F}_p\) denotes the field of cardinality \(p\), a prime number. This result is clearly best possible for groups of size close to \(p^{2/3}\). In the present paper, the authors improve the result for groups of smaller cardinality, answering a question of \textit{T. Cochrane} and \textit{C. Pinner} [Integers 8, No. 1, A46, 18 p. (2008; Zbl 1205.11108)].NEWLINENEWLINEIt is shown that for any \(\varepsilon > 0\) and sufficiently large \(p\) one has for \(p^{\varepsilon} \leq |R| \leq p^{2/3 - \varepsilon}\) that \(|R\pm R| \geq |R|^{3/2+\delta}\) where \(\delta > 0\) depends on \(\varepsilon\) only. Indeed, a more precise result is obtained, which is too technical to recall here.